$\delta$-$J$-ideals of commutative rings

Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $\delta$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $\delta$-$J$-ideal of a commutative ring is defined, that is, if $a,b\in R$ and $ab\in I\in $\mathcal{I}$®$,then$a\in J®$(theJacobsonradicalof$R$)or$b\in \delta(I)$\mathrm{.}Moreover,somepropertiesof$\delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.