A note on weak \(w\)-projective modules (Q6563066)

Let \(R\) a commutative ring and \(M\) an \(R\)-module. Then \(M\) is said to be a weak \(w\)-projective module if \(\mathrm{Ext}^{1}_{R}(M,N) = 0\) for all \(GV\)-torsion-free \(R\)-modules \(N\) with the property that \(\mathrm{Ext}^{k}_{R}(T,N) = 0\) for all \(w\)-projective \(R\)-modules \(T\) and all integers \(k\geq 1\). In the paper under review, the author studied some properties of certain classes of commutative rings using the notion of weak \(w\)-projective modules. Among others, he proved that a ring \(R\) is a \(DW\)-ring (i. e., every ideal is a \(w\)-ideal) if and only if every weak \(w\)-projective module is projective; a ring \(R\) is a von Neumann regular ring if and only if every \(FP\)-projective module is weak \(w\)-projective if and only if every finitely presented \(R\)-module is weak \(w\)-projective; and a ring \(R\) is \(w\)-semi-hereditary if and only if every finite type submodule of a free module is weak \(w\)-projective if and only if every finitely generated ideal of \(R\) is weak \(w\)-projective.