Nil-prime ideals of a commutative ring

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P of R is a nil-prime ideal if there exists x ∈ N(R) and whenever ab ∈ P, then a ∈ P or b ∈ P or a+x ∈ P or b+x ∈ P for each a,b ∈ R. Also, we introduce nil versions of some algebraic concepts in ring theory such as nil-maximal ideal, nil-minimal ideal, nil-principal ideal and investigate some nil-version of a well-known results about them.