On weakly delta-semiprimary ideals of commutative rings

Let $R$ be a commutative ring with $1\mathrm{\ne }0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$,then$a\in \sqrt{I}$or$b\in \sqrt{I}$\mathrm{.}Wesay$I$ is a {\it weakly semiprimary ideal} of $R$ if whenever $a,b\in R$ and $0≠ab\in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let $I(R)$ be the set of all ideals of $R$ and let $\delta :I(R)\to I(R)$ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L,I,J$ are ideals of $R$ with $J\subseteq I$, then $L \subseteq \delta(L)$and$\delta(J) \subseteq \delta(I)$\mathrm{.}Let$\delta$ be an expansion function of ideals of $R$. Then a proper ideal $I$ of $R$ (i.e., $I≠R$) is called a ({\it $\delta$-semiprimary}) {\it weakly $\delta$-semiprimary} ideal of $R$ if ($ab\in I$) $0≠ab\in I$ implies $a\in \delta (I)$ or $b\in \delta (I)$. For example, let $\delta :I(R)\to I(R)$ such that $\delta (I)=\sqrt{I}$. Then $\delta$ is an expansion function of ideals of $R$ and hence a proper ideal $I$ of $R$ is a ($\delta$-semiprimary) weakly $\delta$-semiprimary ideal of $R$ if and only if $I$ is a (semiprimary) weakly semiprimary ideal of $R$. A number of results concerning weakly $\delta$-semiprimary ideals and examples of weakly $\delta$-semiprimary ideals are given.