### $(\delta ,2)$-primary ideals of a commutative ring

Let $R$ be a commutative ring with nonzero identity, let $\mathcal{I}\left(\mathcal{R}\right)$ be the set of all ideals of $R$ and $\delta :\mathcal{I}\left(\mathcal{R}\right)\to \mathcal{I}\left(\mathcal{R}\right)$ an expansion of ideals of $R$ defined by $I\mapsto \delta \left(I\right)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then ${a}^{2}\in I$ or ${b}^{2}\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta $-primary and...