A note on generalizations of semisimple modules

Kaynar, Engin, Türkmen, Burcu N., and Türkmen, Ergül. "A note on generalizations of semisimple modules." Commentationes Mathematicae Universitatis Carolinae 60.3 (2019): 305-312. <http://eudml.org/doc/294249>.

@article{Kaynar2019,
abstract = {A left module $M$ over an arbitrary ring is called an $\mathcal \{RD\}$-module (or an $\mathcal \{RS\}$-module) if every submodule $N$ of $M$ with $\{\rm Rad\}(M)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathcal \{RD\}$-modules and $\mathcal \{RS\}$-modules. We prove that $M$ is an $\mathcal \{RD\}$-module if and only if $M=\{\rm Rad\}(M)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathcal \{RS\}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathcal \{RS\}$-modules. We completely determine the structure of these modules over Dedekind domains.},
author = {Kaynar, Engin, Türkmen, Burcu N., Türkmen, Ergül},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {radical; supplement},
language = {eng},
number = {3},
pages = {305-312},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on generalizations of semisimple modules},
url = {http://eudml.org/doc/294249},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Kaynar, Engin
AU - Türkmen, Burcu N.
AU - Türkmen, Ergül
TI - A note on generalizations of semisimple modules
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 3
SP - 305
EP - 312
AB - A left module $M$ over an arbitrary ring is called an $\mathcal {RD}$-module (or an $\mathcal {RS}$-module) if every submodule $N$ of $M$ with ${\rm Rad}(M)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathcal {RD}$-modules and $\mathcal {RS}$-modules. We prove that $M$ is an $\mathcal {RD}$-module if and only if $M={\rm Rad}(M)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathcal {RS}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathcal {RS}$-modules. We completely determine the structure of these modules over Dedekind domains.
LA - eng
KW - radical; supplement
UR - http://eudml.org/doc/294249
ER -