Nonnil-P-coherent rings and nonnil-PP-rings (Q6654996)

In this paper, all the rings considered are commutative with nonzero identity. Let \(R\) be a ring and \(M\) an \(R\)-module. Let's recall the definitions of the tools used in the paper under review. The module \(M\) is said to be \(\phi-P\)-flat if for each \(s\in R\setminus \mathrm{Nil}(R)\) and each \(x\in M\) stisfying \(sx=0\), then \(x\in (0:s)M\). The module \(M\) is said to be nonnil-\(P\)-injective if for each \(a\in R\setminus \mathrm{Nil}(R)\), every homomorphism from \(Ra\longrightarrow M\) can be extended to a homomorphism from \(R\longrightarrow M\). Also, the ring \(R\) is said to be nonnil-\(P\)-coherent (resp. \(\phi-PF\), nonnil-\(PP\)) if for each \(a\in R\setminus \mathrm{Nil}(R)\), \(Ra\) is a finitely presented (resp. flat, projective) module. In this paper, the authors use the \(\phi-P\)-flat and nonnil-\(P\)-injective modules to investigate the notions of nonnil-\(P\)-coherent rings, \(\phi-PF\)-rings and nonnil \(PP\)-rings.