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

Abstract: Let $R$ be a commutative ring with $1\neq0$. 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 $f:A\rightarrow B$ 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.