Nil-primary ideals of a commutative ring

Let R be a commutative ring with identity and Nil(R) be the set of all nilpotent elements of R. In this paper, we introduce the notion of nil-primary ideals as a generalization of primary ideals and we investigate some nil-versions of the well-known results about primary ideals. We say that a proper ideal P of R is a nil-primary ideal if there exists \(x \in Nil(R)\) $x\in Nil(R)$ such that whenever \(ab \in P\) $ab\in P$ for some \(a, b \in R\) $a,b\in R$ , then \(a\in P\) $a\in P$ or \(b^n \in P\) $b^n\in P$ or \(a+x \in P\) $a+x\in P$ or \(b^n+x \in P\) $b^n+x\in P$ for some \({n \in \mathbb N}\) $n\in \mathbb{N}$ .