On weakly $(m,n)-$closed $\delta -$primary ideals of commutative rings

Abstract

Let $R$ be a commutative ring with $1\ne 0$. In this article, we introduce the concept of weakly $(m,n)-$closed $\delta -$primary ideals of $R$ and explore its basic properties. We show that $I{\bowtie }^{f}J$ is a weakly $(m,n)-$closed ${\delta }_{{\bowtie }^f}-$primary ideal of $A{\bowtie }^{f}J$ that is not $(m,n)-$closed ${\delta }_{{\bowtie }^f}-$primary if and only if $I$ is a weakly $(m,n)-$closed $\delta -$primary ideal of $A$ that is not $(m,n)-$closed $\delta -$primary and for every $\delta$-$(m,n)$-unbreakable-zero element $a$ of $I$ we have $(f(a)+j{)}^m=0$ for every $j\in J$, where $f:A\to B$ is a homomorphism of rings and $J$ is an ideal of $B.$ Furthermore, we provide examples to demonstrate the validity and applicability of our results.