Yazd University Algebraic Structures and Their Applications 2382-9761 11 4 2024 11 01 On left weakly jointly prime $(R,S)$-modules 273 285 3382 10.22034/as.2024.19918.1630 EN Dian Ariesta Yuwaningsih Department of Mathematics Education, Universitas Ahmad Dahlan, Yogyakarta, Indonesia. 0000-0003-0547-9682 Indah Emilia Wijayanti Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia. 0000-0003-0390-8682 Journal Article 2023 04 01 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.<br />  https://as.yazd.ac.ir/article_3382_b95bc4e625f2024c72e2c0eaff4b119d.pdf