On max-flat and max-cotorsion modules

Alagöz, Yusuf; Büyükaşık, Engı̇n

doi:10.1007/s00200-020-00482-4

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…Proof. (1). For any min-pure injective absolutely min-pure right R-module A, By Proposition 2.8, there is a min-pure sequence 0 → A → E → B → 0 with E injective.…”

Section: Corollary 210 the Next Conditions Are True For Any Ring Rmentioning

confidence: 96%

“…Proof. (1). Using the properties of the Ext functor, closedness of absolutely min-purity under extensions is obvious by Proposition 2.8.…”

Section: Some (Pre)envelopes and (Pre)coversmentioning

confidence: 96%

“…Proof. (1). If we assume that every min-pure exact sequence is pure, then every RD-exact sequence is pure, whence R is an RD-ring.…”

Section: Lemma 27 ([30 Proposition 12]) For a Module A R We Havementioning

confidence: 99%

“…In [28], Mao introduced the concept of min-purity and min-pure injectivity, to give further homologic characterizations of min-injective modules and to investigate the existence of min-injective covers. In the literature, purity has a considerable impact on module-ring theory, and several crucial generalizations of this notion are given since it was firstly introduced (see, [1,5,6,8,10,31,35,36]). In accordance with the terminology of Mao [28], a sequence 0 → D → E → F → 0 of right R-modules is called min-pure exact if Hom(R/aR, E) → Hom(R/aR, F ) → 0 is epic for any a ∈ R such that Ra is simple.…”

Section: Introductionmentioning

confidence: 99%

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Homological objects of min-pure exact sequences

Alagöz

Moradzadeh-Dehkordi

2024

Hacettepe Journal of Mathematics and Statistics

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In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring R is two-sided Kothe if all right R-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with J(R)^2 = 0, it is proven that all R-modules are min-pure projective if and only if R is either a ﬁeld or a quasi-Frobenius ring of composition length 2.

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“…Proof. (1). For any min-pure injective absolutely min-pure right R-module A, By Proposition 2.8, there is a min-pure sequence 0 → A → E → B → 0 with E injective.…”

Section: Corollary 210 the Next Conditions Are True For Any Ring Rmentioning

confidence: 96%

“…Proof. (1). Using the properties of the Ext functor, closedness of absolutely min-purity under extensions is obvious by Proposition 2.8.…”

Section: Some (Pre)envelopes and (Pre)coversmentioning

confidence: 96%

“…Proof. (1). If we assume that every min-pure exact sequence is pure, then every RD-exact sequence is pure, whence R is an RD-ring.…”

Section: Lemma 27 ([30 Proposition 12]) For a Module A R We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Homological objects of min-pure exact sequences

Alagöz

Moradzadeh-Dehkordi

2024

Hacettepe Journal of Mathematics and Statistics

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“…Recently, there is a significant interest in some classes of modules that are defined via closed submodules, neat submodules and C-pure submodules (see [1,4,6,23]). A right R-module A is called weakly-flat [23](resp.…”

Section: Introductionmentioning

confidence: 99%

C-Pure Submodules and C-Flat Modules

Alagöz

2021

Journal of Universal Mathematics

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Let R be a ring. A right R-module A is said to be C-flat if the kernel of any epimorphism B → A is C-pure in B, i.e. the induced map Hom(C,B) → Hom(C,A) is surjective for any cyclic right R-module C. Projective modules are C-flat and C-flat modules are weakly-flat and neat-flat. In this article, it is discussed the connections between C-flat, weakly-flat, neat-flat and singly flat modules. It is shown that C-flat modules coincide with singly-projective modules over arbitrary rings. Next, several characterizations of certain classes of rings and modules via C-purity are considered. We prove that, every C-flat module is injective if and only if R is a QF ring. Moreover, we show that R is a CF ring if and only if every FP-injective right R-module is C-flat.

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