On \(\phi\)-Schreier rings (Q2821944)

Let \(R\) be a nonzero commutative unital ring, and let \(N(R)\) denote the nilardical of \(R\), i.e., the set of all nilpotent elements of \(R\). A prime ideal \(P\) of \(R\) is called a \textit{divided prime ideal} if \(P\subseteq (x)\) for any \(x\in R\setminus P\). The authors are interested in the class \(\mathcal H\) of all nonzero commutative unital rings such that \(N(R)\) is a divided prime ideal. They introduce and study the \(\varphi-\mathbb P\) rings, where \(P\) is any of the following classes of rings: Schreier rings, quasi-Schreier rings, almost domains, almost quasi-Schreier rings, GCD rings, generalized GCD rings and almost GCD rings: they say that \(R\in\mathcal H\) is a \(\varphi-\mathbb P\) ring if \(R/N(R)\in\mathbb P\).