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Keywords:
radical; supplement
radical; supplement
Summary:
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.
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