# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 15, 2022

## On Weakly S-prime Submodules

Hani A. Khashan and Ece Yetkin Celikel

Al al-Bayt University, Hasan Kalyoncu University

### Abstract

Let $R$ be a commutative ring with a non-zero identity, $S$ be a
multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this
paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be
weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and
$m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many
properties, examples and characterizations of weakly $S$-prime submodules are
introduced, especially in multiplication modules. Moreover, we investigate the
behavior of this structure under module homomorphisms, localizations, quotient
modules, cartesian product and idealizations. Finally, we define two kinds of
submodules of the amalgamation module along an ideal and investigate
conditions under which they are weakly $S$-prime.

Keywords: S-prime ideal, weakly S-prime ideal, S-prime submodule, weakly S-prime submodule, amalgamated algebra.

MSC numbers: 13A15, 16P40, 16D60.