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Bull. Korean Math. Soc. 2024; 61(3): 717-734

Online first article April 2, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230319

Copyright © The Korean Mathematical Society.

On weakly $(m,n)$-prime ideals of commutative rings

Hani A. Khashan, Ece Yetkin Celikel

Al al-Bayt University; Hasan Kalyoncu University

Abstract

Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. In this paper, we introduce the class of weakly $(m,n)$-prime ideals generalizing $(m,n)$-prime and weakly $(m,n)$-closed ideals. A proper ideal $I$ of $R$ is called weakly $(m,n)$-prime if for $a,b\in R$, $0\neq a^{m}b\in I$ implies either $a^{n}\in I$ or $b\in I$. We justify several properties and characterizations of weakly $(m,n)$-prime ideals with many supporting examples. Furthermore, we investigate weakly $(m,n)$-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.

Keywords: Weakly $(m,n)$-prime ideal, weakly $(m,n)$-closed ideal, $(m,n)$-prime ideal, weakly $n$-absorbing ideal

MSC numbers: Primary 13A15; Secondary 13F05

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