On Left Weakly Jointly Prime $(R,S)$-Modules

Document Type : Research Paper

Authors

1 Department of Mathematics Education, Universitas Ahmad Dahlan, Yogyakarta, Indonesia.

2 Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia.

Abstract

Let $R$ and $S$ be commutative rings and $M$ an $(R,S)$-module. A proper $(R,S)$-submodule $P$ of $M$ is called left weakly jointly prime if for each $(R,S)$-submodule $N$ of $M$ and elements $a,b$ of $R$ such that $abNS\subseteq P$ implies either $aNS\subseteq P$ or $bNS\subseteq P$. This paper defines left weakly jointly prime $(R,S)$-modules and presents some of their properties. On the other hand, a ring $R$ is called fully prime if each proper ideal of $R$ is prime. We extend this fact to $(R,S)$-modules. An $(R,S)$-module $M$ is called fully left weakly jointly prime if each proper $(R,S)$-submodule of $M$ is left weakly jointly prime. Moreover, we present some properties of fully left weakly jointly prime $(R,S)$-modules. At the end of this paper, we present our main results about the necessary and sufficient conditions for an arbitrary $(R,S)$-module to be fully left weakly jointly prime.
 

Keywords

References

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Algebraic Structures and Their Applications

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Available Online from 01 May 2024

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