On the min-projective modules (Q2920225)

Let \(R\) be a commutative ring. An \(R\)-module \(C\) is called cotorsion if \(\mathrm{Ext}^{1}_{R}(F,C)=0\), for all flat \(R\)-module \(F\). An \(R\)-module \(N\) is called finitely presented if there exists an exact sequence \(R^{(n)}\rightarrow R^{(m)}\rightarrow N\rightarrow O\) and an \(R\)-module \(M\) is called \(FP\)-injective if \(\mathrm{Ext}^{1}_{R}(N,M)=0\) for any finitely presented \(R\)-module \(N\). Also an \(R\)-module \(N\) is called \(FP\)-projective if \(\mathrm{Ext}^{1}_{R}(N,M)=0\) for any \(FP\)-injective \(R\)-module \(M\).NEWLINENEWLINEAn \(R\)-module \(M\) is called min-projective if \(\mathrm{Ext}^{1}_{R}(M,\frac{R}{I})=0\) and is called min-flat if \(\mathrm{Tor}^{R}_{1}(N,\frac{R}{I})=0\) for any simple ideal I. If every \(R\)-module \(N\) is min-projective or min-flat then \(R\) is called universally min-projective ring or universally min-flat ring.NEWLINENEWLINEIn this paper, the authors show that a min-flat \(R\)-module \(M\) is min-projective if and only if the \(\frac{R}{I}\)-module \(\frac{M}{MI}\) is min-projective, for every simple ideal I. They give some characterization of min-projective \(R\)-modules on cotorsion rings, von Neumann regular rings and coherent rings, universally min-projective rings and perfect rings. The authors show that \(R\) is a cotorsion ring if and only if every flat \(R\)-module is min-projective; \(R\) is a von Neumann regular ring if and only if \(R\) is a coherent ring and every \(FP\)-projective \(R\)-module is min-projective.NEWLINENEWLINEMoreover, the authors prove that \(R\) is a perfect ring if and only if every min-projective \(R\)-module is cotorsion if and only if every flat \(R\)-module is min-projective and every min-projective \(R\)-module has a cotorsion envelope with the unique mapping property if and only if for each \(R\)-homomorphism \(f:M_{1}\rightarrow M_{2}\) with \(M_{1}\) and \(M_{2}\) min-projective, \(ker(f)\) is cotorsion.