# Communications of theKorean Mathematical SocietyCKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

## Article

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Commun. Korean Math. Soc. 2024; 39(3): 575-583

Online first article July 19, 2024      Printed July 31, 2024

https://doi.org/10.4134/CKMS.c230266

## $(m,n)$-closed $\delta$-primary ideals in amalgamation

Al-Aqsa University; CRMEF Rabat-Sal\'e-Kenitra:

### Abstract

Let $R$ be a commutative ring with $1\neq 0.$ Let $Id(R)$ be the set of all ideals of $R$ and let $\delta: Id(R)\longrightarrow Id(R)$ be a function. Then $\delta$ is called an expansion function of the ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J\subseteq I,$ then $L\subseteq \delta(L)$ and $\delta(J)\subseteq \delta(I).$ Let $\delta$ be an expansion function of the ideals of $R$ and $m\geq n>0$ be positive integers. Then a proper ideal $I$ of $R$ is called an \textit{$(m,n)$-closed $\delta$-primary ideal} (resp., \textit{weakly $(m,n)$-closed $\delta$-primary ideal}) if $a^{m}\in I$ for some $a\in R$ implies $a^{n}\in\delta(I)$ (resp., if $0\neq a^{m}\in I$ for some $a\in R$ implies $a^{n}\in\delta(I)).$ Let $f:A\longrightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B.$ This paper investigates the concept of $(m,n)$-closed $\delta$-primary ideals in the amalgamation of $A$ with $B$ along $J$ with respect to $f$ denoted by $A\bowtie^{f}J.$

Keywords: $\delta$-primary ideal, $(m,n)$-closed $\delta$-primary ideal, weakly $(m,n)$-closed $\delta$-primary ideal, Amalgamation, trivial extension

MSC numbers: Primary 13E015, 13E10, 13E99

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