The torsion theory generated by M-small modules

A. Çiǧdem Özcan; Abdullah Harmanci

doi:10.1007/s100110300006

The torsion theory generated by M-small modules

A. Çiǧdem Özcan
, Abdullah Harmanci

Division of Algebra and Number Theory

Hacettepe University

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Let M be a right R-module, ℳ the class of all M-small modules, and P a projective cover of M in σ[M]. We consider the torsion theories τ_ℳ = (τ_ℳ, ℱ_ℳ), τ_V = (τ_V,ℱ_V), and τ _P = (τ_P, ℱ_P) in σ[M], where τ_ℳ is the torsion theory generated by ℳ, τ _V is the torsion theory cogenerated by ℳ, and τ_P is the dual Lambek torsion theory. We study some conditions for τ _ℳ to be cohereditary, stable, or split, and prove that Rej(M, ℳ) = M ⇔ ℱ_P = ℳ (= τ_ℳ = ℱ_V) ⇔ τ_P = τ_V ⇔ Gen _M(P) ⊆ τ_V.

Original language	English
Pages (from-to)	41-52
Number of pages	12
Journal	Algebra Colloquium
Volume	10
Issue number	1
DOIs	https://doi.org/10.1007/s100110300006
Publication status	Published - Mar 2003

Keywords

Hereditary torsion theory
Small module

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10.1007/s100110300006

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abstract = "Let M be a right R-module, ℳ the class of all M-small modules, and P a projective cover of M in σ[M]. We consider the torsion theories τℳ = (τℳ, ℱℳ), τV = (τV,ℱV), and τ P = (τP, ℱP) in σ[M], where τℳ is the torsion theory generated by ℳ, τ V is the torsion theory cogenerated by ℳ, and τP is the dual Lambek torsion theory. We study some conditions for τ ℳ to be cohereditary, stable, or split, and prove that Rej(M, ℳ) = M ⇔ ℱP = ℳ (= τℳ = ℱV) ⇔ τP = τV ⇔ Gen M(P) ⊆ τV.",

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TY - JOUR

T1 - The torsion theory generated by M-small modules

AU - Özcan, A. Çiǧdem

AU - Harmanci, Abdullah

PY - 2003/3

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N2 - Let M be a right R-module, ℳ the class of all M-small modules, and P a projective cover of M in σ[M]. We consider the torsion theories τℳ = (τℳ, ℱℳ), τV = (τV,ℱV), and τ P = (τP, ℱP) in σ[M], where τℳ is the torsion theory generated by ℳ, τ V is the torsion theory cogenerated by ℳ, and τP is the dual Lambek torsion theory. We study some conditions for τ ℳ to be cohereditary, stable, or split, and prove that Rej(M, ℳ) = M ⇔ ℱP = ℳ (= τℳ = ℱV) ⇔ τP = τV ⇔ Gen M(P) ⊆ τV.

AB - Let M be a right R-module, ℳ the class of all M-small modules, and P a projective cover of M in σ[M]. We consider the torsion theories τℳ = (τℳ, ℱℳ), τV = (τV,ℱV), and τ P = (τP, ℱP) in σ[M], where τℳ is the torsion theory generated by ℳ, τ V is the torsion theory cogenerated by ℳ, and τP is the dual Lambek torsion theory. We study some conditions for τ ℳ to be cohereditary, stable, or split, and prove that Rej(M, ℳ) = M ⇔ ℱP = ℳ (= τℳ = ℱV) ⇔ τP = τV ⇔ Gen M(P) ⊆ τV.

KW - Hereditary torsion theory

KW - Small module

UR - https://www.scopus.com/pages/publications/0042525703

U2 - 10.1007/s100110300006

DO - 10.1007/s100110300006

M3 - Article

AN - SCOPUS:0042525703

SN - 1005-3867

VL - 10

SP - 41

EP - 52

JO - Algebra Colloquium

JF - Algebra Colloquium

IS - 1

ER -

The torsion theory generated by M-small modules

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