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Title: Precovers and Goldie’s torsion theory (English)
Author: Bican, Ladislav
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 128
Issue: 4
Year: 2003
Pages: 395-400
Summary lang: English
Category: math
Summary: Recently, Rim and Teply , using the notion of $\tau $-exact modules, found a necessary condition for the existence of $\tau $-torsionfree covers with respect to a given hereditary torsion theory $\tau $ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau $-torsionfree and $\tau $-exact covers have been investigated in . The purpose of this note is to show that if $\sigma = (\mathcal T_{\sigma },\mathcal F_{\sigma })$ is Goldie’s torsion theory and $\mathcal F_{\sigma }$ is a precover class, then $\mathcal F_{\tau }$ is a precover class whenever $\tau \ge \sigma $. Further, it is shown that $\mathcal F_{\sigma }$ is a cover class if and only if $\sigma $ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that $\mathcal F_{\tau }$ is a cover class for all hereditary torsion theories $\tau \ge \sigma $. (English)
Keyword: hereditary torsion theory
Keyword: Goldie’s torsion theory
Keyword: non-singular ring
Keyword: precover class
Keyword: cover class
Keyword: torsionfree covers
Keyword: lattices of torsion theories
MSC: 16D80
MSC: 16D90
MSC: 16S90
MSC: 18E40
idZBL: Zbl 1057.16027
idMR: MR2032476
DOI: 10.21136/MB.2003.134006
Date available: 2009-09-24T22:11:09Z
Last updated: 2020-07-29
Stable URL:
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