diff --git "a/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt" "b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt" @@ -0,0 +1,577 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf,len=576 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='00279v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='AC] 31 Dec 2022 A NOTE ON WEAK w-PROJECTIVE MODULES REFAT ABDELMAWLA KHALED ASSAAD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' An R-module M is said to be a weak w-projective module if Ext1 R(M, N) = 0 for all N ∈ P†∞ w (see, [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In this paper, we in- troduce and study some properties of weak w-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' And we use these modules to characterize some classical rings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' we will prove that a ring R is a DW -ring if and only if every weak w-projective is projective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' R is a Von Neumann regular ring if and only if every FP-projective is weak w- projective if and only if every finitely presented R-module is weak w-projective and R is a w-semi-hereditary if and only if every finite type submodule of a free module is weak w-projective if and only if every finitely generated ideal of R is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Introduction In this paper, all rings are considered commutative with unity and all modules are unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring and M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' As usual, we use pdR(M), idR(M) and fdR(M) to denote, respectively, the classical projective dimension, injective dimension and flat dimension of M, and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) and gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) to denote, respectively, the weak and global homological dimensions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Now, we review some definitions and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [23], J is called a Glaz-Vasconcelos ideal (a GV -ideal for short) if J is finitely gener- ated and the natural homomorphism ϕ : R → J∗ = HomR(J, R) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Note that the set GV (R) of GV -ideals of R is a multiplicative system of ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is Defined torGV (M) = {x ∈ M | Jx = 0 for some J ∈ GV (R)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is clear that torGV (M) is submodule of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' M is said to be GV -torsion (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', GV -torsion-free) if torGV (M) = M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', torGV (M) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A GV -torsion-free module M is called a w-module if Ext1 R(R/J, M) = 0 for any J ∈ GV (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, projective modules and reflexive modules are w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the recent paper [23], it was shown that flat modules are w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Also it is known that a GV -torsion- free R-module M is a w-module if and only ExtR 1 (N, M) = 0 for every GV -torsion R-module N (see, [13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The notion of w-modules was introduced firstly over a domain [17] in the study of Strong Mori domains and was extended to commutative rings with zero divisors in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let w − Max(R) denote the set of w-ideals of R maximal among proper integral w-ideals of R (maximal w-ideals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [23, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8], every maximal w-ideal is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any GV -torsion free module M, Mw := {x ∈ E(M) | Jx ⊆ M for some J ∈ GV (R)} 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 13D05, 13D07, 13H05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' projective modules , weak w-projective modules, w-flat, GV -torsion, finitely presented type, DW -rings, coherent rings, w-coherent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 1 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD is a w-submodule of E(M) containing M and is called the w-envelope of M, where E(M) denotes the injective hull of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is clear that a GV -torsion-free module M is a w-module if and only if Mw = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M and N be R-modules and let f : M → N be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [12], f is called a w-monomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', w-epimorphism, w-isomorphism) if fm : Mm → Nm is a monomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', an epimorphism, an isomorphism) for all m ∈ w − Max(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A sequence A → B → C of modules and homomorphisms is called w-exact if the sequence Am → Bm → Cm is exact for all m ∈ w − Max(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' An R-module M is said to be of finite type if there exists a finitely generated free R-module F and a w-epimorphism g : F → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Similarly, an R-module M is said to be of finitely presented type if there exists a w-exact sequence F1 → F0 → M → 0, where F1 and F0 are finitely generated free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In recent years, homological theoretic characterization of w-modules has received attention in several papers the literature (for example see [[1], [21], [19], [18]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The notion of w-projective modules and w-flat modules appeared first in [11] when R is an integral domain and was extended to an arbitrary commutative ring in [[14], [2]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In [14], F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Wang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Kim generalized projective modules to w-projective modules by the w-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' An R-module M is said to be a w- projevtive if Ext1 R(L(M), N) is GV -torsion for any torsion-free w-module N, where L(M) = (M/ torGV (M))w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Pw the class of all w-projective R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [20], an R-module M is a w-split if and only if Ext1 R(M, N) is GV -torsion for all R-modules N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Sw the class of all w-split R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4], every w-split module is w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [2], an R- module M is said to be w-flat if for any w-monomorphism f : A → B, the induced sequence 1 ⊗ f : M ⊗R A → M ⊗R B is a w-monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Fw the class of all w-flat R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [18], throughout this paper, P†∞ w denote the class of GV -torsion-free R-modules N with the property that Extk R(M, N) = 0 for all w-projective R-modules M and for all integers k ≥ 1 Clearly, every GV - torsionfree injective R-module belongs to P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' An R-module M is said to be weak w-projective if Ext1 R(M, N) = 0 for all N ∈ P†∞ w : Denote by wPw the class of all weak w-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [18], Wang and Qiao introduce the notions of the weak w-projective dimension (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pd) of a module and the global weak w-projective dimension (gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-dim) of a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [18], a GV -torsion- free module M is said to be a strong w-module if Exti R(N, M) = 0for any integer i ≥ 1 and all GV -torsion modules N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by W∞ the class of all strong w- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then all GV -torsion-free injective modules are strong w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Clearly, P†∞ w ⊆ W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' But, in [18], they do not showed that P†∞ w and W∞ are the different class of R-modules, and this question was answered in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Recall from [4] that an R-module M is called FP-projective if Ext1 R(M, N) = 0 for any absolutely pure R-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by FP the class of all FP-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Recall that an R-module A is an absolutely pure if A is a pure submodule in every R-module which contains A as a submodule (see, [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Megibbeni showed in [5], that an R-module A is absolutely pure if and only if Ext1 R(F, A) = 0, for every finitely presented module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, an absolutely pure module is precisely a FP-injective module in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A NOTE ON WEAK w-PROJECTIVE MODULES 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' results In this section, we introduce a characterize of some classical ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' But we need the following lemma Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ([18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5) An R-module M is weak w-projective if Ext1 R(M, N) = 0 for all N ∈ P†∞ w and for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is obvious that, for the class of modules { projective } ⊆ { w-split } ⊆ { w-proective } ⊆ {weak w-proective } ⊆ { w-flat }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[1], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], if R is a perfect ring, then the five classes of modules above coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the following proposition, we will give some characterizations of weak w- projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then the following are equivalent: (1) M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M ⊗ F is weak w-projective for any projective R-module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) HomR(F, M) is weak w-projective for any finitely generated projective R- module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) For any exact sequence of R-modules 0 → A → B → C → 0 with A ∈ P†∞ w , the sequence 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) For any w-exact sequence of R-modules 0 → L → E → M → 0 the se- quence 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0 is exact for any R-module N ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) For any exact sequence of R-modules 0 → L → E → M → 0 the sequence 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0 is exact for any R-module N ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let F be a projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any R-module N in P†∞ w , we have Ext1 R(F ⊗M, N) ∼= HomR(F, Ext1 R(M, N)) by [[13], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is a weak w-projective, Ext1 R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, Ext1 R(F ⊗ M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, F ⊗ M is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1) and (3) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Follow by letting F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N ∈ P†∞ w , for any finitely generated projective R-module F, we have F ⊗ Ext1 R(M, N) ∼= Ext1 R(HomR(F, M), N) by [[13], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective, so Ext1 R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Ext1 R(HomR(F, M), N) = 0, which implies that HomR(F, M) is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → A → B → C → 0 be an exact sequence with A ∈ P†∞ w , then we have the exact sequence 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → Ext1 R(M, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective and A ∈ P†∞ w , so Ext1 R(M, A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N ∈ P†∞ w , consdier an exact sequence 0 → N → E → L → 0 with E is injective module, then we have the exact sequence 0 → HomR(M, N) → HomR(M, E) → HomR(M, L) → Ext1 R(M, N) → 0, and keeping in mind that 0 → HomR(M, N) → HomR(M, E) → HomR(M, L) → 0 is exact, we deduce that Ext1 R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → L → E → M → 0 be a w-exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any R- module N ∈ P†∞ w so N ∈ W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1], we have the exact sequence 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → Ext1 R(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective, so Ext1 R(M, N) = 0, and (5) is holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → L → E → M → 0 be an exact sequence with E is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, for any R-module N ∈ P†∞ w , we have 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → Ext1 R(M, N) → 0 is exact sequence, and keeping in mind that 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0 is exact, we deduce that Ext1 R(M, N) = 0, which implies that M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall from [12], that a ring is said to be w-coherent if every finitely generated ideal of R is of finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a w-coherent ring, E be an injective R-module, M be a finitely presented type and N be an R{x}-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, if M is weak w-projective R-module, so TorR n (M, Hom(N, E)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a weak w-projective R-module and let N be an R{x}-modul, so Extn R(M, n) = 0 by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5] and since every R{x}-module in P†∞ w by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Henace, by [[16], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='13(6)], we have TorR n (M, Hom(N, E)) ∼= Hom(Extn R(M, N), E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Which implies that, TorR n (M, Hom(N, E)) = 0 □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Every weak w-projective of finite type is of finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a weak w-projective R-module of finite type, so by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9] M is w-projective of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='22], we have M is finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) Mw/M is a weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M is a weak w-projective if and only if so is Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' So, by [[13], Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5] we have Mw/M a GV -torsion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3(2)], we have Mw/M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N be an R-module in P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is GV -torsion-free, we have by (1) Mw/M is weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Consider the following exact sequence 0 → M → Mw → Mw/M → 0 which is w-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], M is weak w-projective if and only if Mw is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall that a ring R is called a DW-ring if every ideal of R is a w-ideal, or equivalently every maximal ideal of R is w-ideal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Examples of DW-rings are Pr¨ufer domains, domains with Krull dimension one, and rings with Krull dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' We note that if R is DW-ring, then every R-module in P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the following proposition, we will give a new characterizations of DW-rings which are the only rings with these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following statements are equivalent: A NOTE ON WEAK w-PROJECTIVE MODULES 5 (1) Every weak w-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every w-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every GV -torsion R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) Every GV -torsion-free R-module is strong w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) Every finitely presented type w-flat is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) Every weak w-projective R-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (7) R is DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2) and (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free R-module, for any GV -torsion R-module N, we have Exti R(N, M) = 0 since N is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a strong w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[12], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8] since every strong w-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial, since every projective R-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented type w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9], we have M is a finite type w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a projective R-module by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, M is finitely presented type w-flat, so M is projective by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[9], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1], we have R is a DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5, we have Mw/M is weak w-projectiveis and so w-module by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, Mw/M is a GV -torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Mw/M = 0 and so Mw = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, M is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, R is a DW-ring by [[12], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (7) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any R-module N, we have Ext1 R(M, N) = 0 because N ∈ P†∞ w (since R is DW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Note that the equivalence (1) ⇔ (7) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6 was given in [[8], Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4] for the domain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Mao and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Ding in [[4]], proved that a ring R is a Von Neumann regular if and only if every FP-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Next, we will give new characterizations of a Von Neumann regular rings by weak w-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the following statements are equivalent: (1) Every FP-projective R-module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every finitely presented R-module is weak w-projectiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every finitely presented R-module is w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) R is a Von Neumann regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Follows from the fact that every finitely presented R-module is FP-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented R-module, so M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let I be a finitely generated ideal of R, then R/I is finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' So R/I is w-flat by (3), then w − fdR(R/I) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, w − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) = 0 by [[19], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, R is Von Neumann regular by [[15], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a FP-projective, so M is projective by [[4], Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Next, we will give an example of FP-projective module which is not weak w- projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Consider the local Quasi-Frobenius ring R := k[X]/(X2) where k is a field, and denote by X the residue class in R of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, (X) is FP-projective R-module which is not weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since R is a Quasi-Frobenius ring, then every absolutely pure R-module is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, for any absolutely pure R-module N, we have Ext1 R((X), N) = 0, so (X) is FP-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' But, (X) is not projective by [[10], Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2], and so not weak w-projective, since R is DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall from [[15]] that a ring R is said to be w-semi-hereditary if every finite type ideal of R is w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent: (1) R w-semi-hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every finite type submodule of a free module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every finite type ideal of R is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) Every finitely generated submodule of a free module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) Every finitely generated ideal of R is a weak w-projective Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be a finite type submodule of a any free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, J is w-projective by [[15], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then J is weak w-projective by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3) ⇒ (5) and (2) ⇒ (4) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' These are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be a finite type ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then J is w-isomorphic to a finitely generated subideal I of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence J is weak w-projective by hypothesis and [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Every GV -torsion-free weak w-projective module is torsion- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a GV - torsion-free w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='Thus, by [[13], Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6], we have M is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ In the next example we will prove that a weak w-projective module need not to be torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be an integral domain and J be a proper GV -ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then M := R ⊕ R/J is a weak w-projective module but not torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring and M be a finitely presented R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the following statements are equivalent: (1) M is w-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) For any w-exact 0 → A → B → C → 0 , the sequence 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is w-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial, since every w-split R-module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → A → B → C → 0 be a w-exact sequence of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, for any maximal w-ideal m of R, 0 → Am → Bm → Cm → 0 is exact sequence of Rm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, since Mm is free by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8], we have the exact A NOTE ON WEAK w-PROJECTIVE MODULES 7 sequence 0 → HomR(Mm, Am) → HomR(Mm, Bm) → HomR(Mm, Cm) → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is finitely presented, we have the commutative diagram HomRm(Mm, Am) → HomRm(Mm, Bm) → HomRm(Mm, Cm) || ≀ || ≀ || ≀ HomR(M, A)m → HomR(M, B)m → HomR(M, C)m Thus, 0 → HomR(M, A)m → HomR(M, B)m → HomR(M, C)m → 0 is exact, and so, 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is w-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[20], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall from [22], that a w-exact sequence of R-modules 0 → A → B → C → 0 is said to be w-pure exact if, for any R-module M, the induced sequence 0 → A ⊗ M → B ⊗ M → C ⊗ M → 0 is w-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let C be a finitely presented type R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the follow- ing statements are equivalent: (1) C is a weak w-projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every w-exact sequence of R-modules 0 → A → B → C → 0 is w-pure exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since every weak w-projective is w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[22], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6], we have the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → A → B → C → 0 be a w-exact sequence, so is a w-pure exact by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, C is w-flat by [[22], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence C is a weak w-projective by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent for a finite type R-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) M is a w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Ext1 R(M, B) = 0 for any B ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Ext1 R(M, N) = 0 for any R{x}-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) M{x} is a projective R{x}-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' This is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N be an R{x}-module, we have by [[16], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], Extn R{x}(M{x}, N) ∼= Extn R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, M{x} is a projective R{x}-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M{x} be a projective R{x}-module, so M{x} is finitely generated by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='24] and since M is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='18], M is w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall form [[18]], that an R-module D is said to be P†∞ w -divisible if it is iso- morphic to E/N where E is a GV -torsin-free injective R-module and N ∈ P†∞ w is a submodule of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module and any integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Extm R (M, D) = 0 for all P†∞ w -divisible R-module D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then there exists an exact sequence of R-modules 0 → N → E → H → 0, where E is a GV -torsion-free injective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, D is P†∞ w -divisibl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then we have the induced exact sequence Extm R (M, H) → Extm+1 R (M, N) → Extm+1 R (M, E) = 0, for any integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The left term is zero by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Extm+1 R (M, N) = 0, which implies that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m and D be a P†∞ w -divisible R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then we have an exact sequence 0 → N → E → H → 0, where E is a GV -torsion-free injective R-module and N ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, we have the exact sequence 0 = Extm R (M, E) → Extm R (M, H) → Extm+1 R (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The right term is zero by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Therefore, Extm R (M, H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M and N be two R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) = sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The inequality w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) ≤ sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} follows from the fact that the class of weak w-projective modules is closed under direct sums by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For the converse inequality, we may assume that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) = n is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, for any R-module X ∈ P†∞ w , Extn+1 R (M ⊕ N, X) ∼= Extn+1 R (M, X) ⊕ Extn+1 R (N, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since Extn+1 R (M ⊕ N, X) = 0 by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Extn+1 R (M, X) = Extn+1 R (N, X) = 0, which implies that, sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Almahdi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Tamekkante and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} 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