Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2022; 59(6): 1387-1408

Online first article July 15, 2022      Printed November 30, 2022

https://doi.org/10.4134/BKMS.b210784

Copyright © The Korean Mathematical Society.

On weakly $S$-prime submodules

Hani A. Khashan, 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 $(N:_{R}M)\cap S=\emptyset$ 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: Primary 13A15, 16P40; Secondary 16D60

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