Extending modules relative to a torsion theory

Doğruöz, Semra. "Extending modules relative to a torsion theory." Czechoslovak Mathematical Journal 58.2 (2008): 381-393. <http://eudml.org/doc/31216>.

@article{Doğruöz2008,
abstract = {An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau $-extending module, where $\tau $ is a hereditary torsion theory on $\mathop \{\text\{Mod\}\}$-$R$. An $R$-module $M$ is called type 2 $\tau $-extending if every type 2 $\tau $-closed submodule of $M$ is a direct summand of $M$. If $\tau _I$ is the torsion theory on $\mathop \{\text\{Mod\}\}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $\tau _I$-extending $R$-module, then the question of whether or not $M/MI$ is an extending $R/I$-module is investigated. In particular, for the Goldie torsion theory $\tau _G$ we give an example of a module that is type 2 $\{\tau \}_G$-extending but not extending.},
author = {Doğruöz, Semra},
journal = {Czechoslovak Mathematical Journal},
keywords = {torsion theory; extending module; closed submodule; hereditary torsion theories; extending modules; closed submodules; direct summands; Goldie torsion theory},
language = {eng},
number = {2},
pages = {381-393},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extending modules relative to a torsion theory},
url = {http://eudml.org/doc/31216},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Doğruöz, Semra
TI - Extending modules relative to a torsion theory
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 381
EP - 393
AB - An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau $-extending module, where $\tau $ is a hereditary torsion theory on $\mathop {\text{Mod}}$-$R$. An $R$-module $M$ is called type 2 $\tau $-extending if every type 2 $\tau $-closed submodule of $M$ is a direct summand of $M$. If $\tau _I$ is the torsion theory on $\mathop {\text{Mod}}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $\tau _I$-extending $R$-module, then the question of whether or not $M/MI$ is an extending $R/I$-module is investigated. In particular, for the Goldie torsion theory $\tau _G$ we give an example of a module that is type 2 ${\tau }_G$-extending but not extending.
LA - eng
KW - torsion theory; extending module; closed submodule; hereditary torsion theories; extending modules; closed submodules; direct summands; Goldie torsion theory
UR - http://eudml.org/doc/31216
ER -