2-nil ideals of commutative rings (Q2078112)

Let \(R\) be a commutative ring with identity and \(I\) an ideal of \(R\). The author proposes a generalization of a prime ideal in the following way. We say that \(I\) is a 2-nil ideal if for all \(a,b,c\in R\), we have \(abc\in R\Rightarrow ab\in\sqrt 0\) or \(ac\in I\) or \(bc\in I\). Then she compares this notion with many other concepts defined in the literature. A special attention is given to rings having the property that prime ideals coincide with 2-nil ideals. Many interesting situations are considered.