Abelian groups

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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“…By (1.9) and the remark after (1.15) in [8] and also by Corollary of Lemma 2.1 in [14] we obtain the following corollary. Proof.…”

Section: Homological Objects Ofsupporting

confidence: 54%

“…Then there is a projective module P and its finitely generated submodule K such that M F/K. Then K is -coinjective by Corollary 4.14, therefore M is -coprojective by (1.12) in [8].…”

Section: Homological Objects Ofmentioning

confidence: 91%

The Proper Class Generated by Weak Supplements

Alizade

Demirci

Durǧun

et al. 2013

Communications in Algebra

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“…We refer the reader to [14,19] for the basic properties of purity and the related notions. We remark that in addition to the term "purity"…”

Section: And+/(~)mentioning

confidence: 99%

“…Let ~) be a (nonhereditary) theory of torsion [14] in Moa ~ such that ~ is the least semisimple class containing ~ [12, p. 369] we have Z~ = -~(~), where ~(P) is the defect of the module [15,12]. Therefore, we can use the properties of the defect to extend the rank function to sod ~ in such a way that the rank becomes an additive function on short exact sequences.…”

Section: Introductionmentioning

confidence: 99%

Algebraically compact modules and relative homological algebra on tame hereditary algebras

Generalov¹

1991

Algebra and Logic

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“…Lifting modules have been generalized in [4] to E-lifting modules by using instead of direct summands (i.e. splitting short exact sequences) elements of a proper class E of short exact sequences in the sense of Buchsbaum [2] or Mishina and Skornjakov [7].…”

Section: Introductionmentioning

confidence: 99%