On \(\varphi \)-strong Mori rings (Q2901914)

A commutative ring \(R\) is called a \(\phi\)-ring if its nilradical \(\mathfrak{n}(R)\) is a divided prime ideal. Several ring notions of the classical multiplicative ideal theory of integral domaind have extended the \(\phi\)-rings like \(\phi\)-noetherian, \(\phi\)-Prüfer, \(\phi\)-Krull and etc. In the paper under review the authors extended the notion of strong Mori domains to the case of \(\phi\)-rings. Let \(R\) be a \(\phi\)-ring with total quotient ring \(T(R)\), and define \(\phi:T(R)\to R_{\mathfrak{n}(R)}\) by \(\phi(a/b)=a/b\). An ideal \(I\) of \(R\) is called a nonnil ideal if it is not contained in \(\mathfrak{n}(R)\). A nonnil ideal \(I\) is called a \(\phi\)-\(w\)-ideal if \(\phi(I)\) is a \(w\)-ideal of \(\phi(R)\). A \(\phi\)-ring is called a \(\phi\)-strong Mori or \(\phi\)-SM ring if it satisfies the ascending chain condition on \(\phi\)-\(w\)-ideals. They proved several results on \(\phi\)-SM rings, one of them is that: Let \(R\) be a \(\phi\)-ring. Then \(R\) is a \(\phi\)-SM ring if and only if \(R/\mathfrak{n}(R)\) is a strong Mori domain.