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.