Generation of proper classes of short exact sequences

Pancar, Ali

doi:10.1155/s016117129700063x

Cited by 6 publications

(10 citation statements)

References 7 publications

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“…We first note that injective and projective objects in this class coincide with the injective and projective objects in mall, , and (see [10]). And then, we prove that is coinjectively generated, namely, it is the smallest proper class for which every small module is coinjective.…”

Section: The Proper Class Generated By Weak Supplements 57mentioning

confidence: 97%

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The Proper Class Generated by Weak Supplements

Alizade

Demirci

Durǧun

et al. 2013

Communications in Algebra

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We show that, for hereditary rings, the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules, submodules that have supplements and weak supplement submodules coincide. Moreover, we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective, projective, coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally, we describe this class for Dedekind domains in terms of supplement submodules.

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Section: The Proper Class Generated By Weak Supplements 57mentioning

confidence: 97%

“…We say that is the proper class generated by (see [10]). Clearly, is the smallest proper class containing .…”

Section: Preliminariesmentioning

confidence: 99%

The Proper Class Generated by Weak Supplements

Alizade

Demirci

Durǧun

et al. 2013

Communications in Algebra

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“…The intersection of all proper classes containing the class P is clearly a proper class, denoted by P . The class P is the smallest proper class containing P, called the proper class generated by P. A module M is called P-projective if it is projective with respect to all short exact sequences in P, that is, Hom(M, E) is exact for every E in P. Notice that the proper class P has the same projective modules as P (see [22]). A module M is called P-coprojective if every short exact sequence of the form 0 → A → B → M → 0 is in P. For a given class M of modules, denote by k(M) the smallest proper class for which each M ∈ M is k(M)-coprojective; it is called the proper class coprojectively generated by M. The largest proper class P for which each M ∈ M is P-projective is called the proper class projectively generated by M. See [25] and [20] for further details on proper classes.…”

Section: P-4)mentioning

confidence: 99%

Proper classes generated by τ- closed submodules

Durǧun

Çobankaya

2019

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

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“…For any class P of short exact sequences the intersection P of all proper classes containing P is clearly a proper class. We say that P is the proper class generated by P (see [15]). Clearly P is the least proper class containing P.…”

Section: Examplementioning

confidence: 99%

Small supplements, Weak supplements and Proper Classes

Alizade¹

2016

HJMS

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Let SS denote the class of short exact sequences E :0 → A f → B → C → 0 of R-modules and R-module homomorphisms such that f (A) has a small supplement in B i.e. there exists a submodule K of M such that f (A) + K = B and f (A) ∩ K is a small module. It is shown that, SS is a proper class over left hereditary rings. Moreover, in this case, the proper class SS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplement submodules. The homological objects, such as, SS-projective and SScoinjective modules are investigated. In order to describe the class SS, we investigate small supplemented modules, i.e. the modules each of whose submodule has a small supplement. Besides proving some closure properties of small supplemented modules, we also give a complete characterization of these modules over Dedekind domains.

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Generation of proper classes of short exact sequences

Cited by 6 publications

References 7 publications

The Proper Class Generated by Weak Supplements

The Proper Class Generated by Weak Supplements

Proper classes generated by τ- closed submodules

Small supplements, Weak supplements and Proper Classes

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