Precovers and Goldie’s torsion theory

Bican, Ladislav. "Precovers and Goldie’s torsion theory." Mathematica Bohemica 128.4 (2003): 395-400. <http://eudml.org/doc/249217>.

@article{Bican2003,
abstract = {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 $.},
author = {Bican, Ladislav},
journal = {Mathematica Bohemica},
keywords = {hereditary torsion theory; Goldie’s torsion theory; non-singular ring; precover class; cover class; torsionfree covers; lattices of torsion theories; hereditary torsion theories; Goldie torsion theory; non-singular rings; precover classes; cover classes; torsionfree covers; lattices of torsion theories},
language = {eng},
number = {4},
pages = {395-400},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Precovers and Goldie’s torsion theory},
url = {http://eudml.org/doc/249217},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Bican, Ladislav
TI - Precovers and Goldie’s torsion theory
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 4
SP - 395
EP - 400
AB - 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 $.
LA - eng
KW - hereditary torsion theory; Goldie’s torsion theory; non-singular ring; precover class; cover class; torsionfree covers; lattices of torsion theories; hereditary torsion theories; Goldie torsion theory; non-singular rings; precover classes; cover classes; torsionfree covers; lattices of torsion theories
UR - http://eudml.org/doc/249217
ER -