Geodesic
A note on generalizations of semisimple modules
Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 3, pp. 305-312.

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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.
DOI : 10.14712/1213-7243.2019.011
Classification : 16D10, 16D99
Keywords: radical; supplement 
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Kaynar, Engin; Türkmen, Burcu N.; Türkmen, Ergül. A note on generalizations of semisimple modules. Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 3, pp. 305-312. doi : 10.14712/1213-7243.2019.011. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2019.011/

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