J-Ideals of Commutative Rings

Let R be a commutative ring with identity and $N(R)$ and J(R) denote the nilradical and the Jacobson radical of R, respectively. A proper ideal I of R is called an n-ideal if for every a,b∈ R, whenever ab∈ I\ and a∉ N(R), then b∈ I. In this paper, we introduce and study J-ideals as a new generalization of n-ideals in commutative rings. A proper ideal I\ of R\ is called a J-ideal if whenever ab∈ I\ with a∉ J(R), then b∈ I\ for every a,b∈ R. We study many properties and examples of such class of ideals. Moreover, we investigate its relation with some other classes of ideals such as r-ideals, prime, primary and maximal ideals. Finally, we, more generally, define and study J-submodules of an R-modules M. We clarify some of their properties especially in the case of multiplication modules.