On strongly nonnil-coherent rings and strongly nonnil-Noetherian rings (Q6563277)

All rings considered in this paper are commutative, unitary and have prime nilradical. Let \(R\) be a ring. Call \(R\) a \(\phi\)-ring if its nilradical Nil\((R)\) is contained in \(xR\) for every non-nilpotent \(x\in R\). Say that a \(\phi\)-ring \(R\) is a nonnil-coherent ring if every finitely generated ideal not contained in Nil\((R)\) is finitely presented. This concept was introduced and studied by \textit{K. Bacem} and \textit{A. Benhissi} [Beitr. Algebra Geom. 57, No. 2, 297--305 (2016; Zbl 1341.13002)]. In the paper under review the authors introduce a subclass of nonnil-coherent rings. They call \(R\) a strongly nonnil-coherent ring if \(R\) is a \(\phi\)-ring and every finitely presented \(\phi\)-torsion \(R\)-module has a (possibly infinite) free resolution consisting of finitely generated modules. Here an \(R\)-module \(M\) is \(\phi\)-torsion if each \(x\in M\) is annihilated by some non-nilpotent scalar. The authors prove many homological results involving the strongly nonnil-coherent rings. Part of these results have classical flavor. For instance, they prove that a \(\phi\)-ring \(R\) is strongly nonnil-coherent iff every direct product of strongly \(\phi\)-flat modules is strongly \(\phi\)-flat. An \(R\)-module \(M\) is called strongly \(\phi\)-flat if Tor\(_n(Q, M) = 0\) for any \(\phi\)-torsion module \(Q\) and any \(n \geq 1\).