ON JOINTLY PRIME RADICALS OF (R,S)-MODULES

Abstract

Let $M$ be an $(R,S)$-module. In this paper a generalization of the m-system set of modules to $(R,S)$-modules is given. Then for an $(R,S)$-submodule $N$ of $M$, we define $\sqrt[(R,S)]{N}$ as the set of $a\in M$ such that every m-system containing $a$ meets $N$. It is shown that $\sqrt[(R,S)]{N}$ is the intersection of all jointly prime $(R,S)$-submodules of $M$ containing $N$. We define jointly prime radicals of an $(R,S)$-module $M$ as $rad_{(R,S)}(M)=\sqrt[(R,S)]{0}$. Then we present some properties of jointly prime radicals of an $(R,S)$-module.DOI : http://dx.doi.org/10.22342/jims.21.1.199.25-34