Relative regular ring and IF ring (Q2747781)

Let \(M\) be a fixed right module over the ring \(R\). Recall that (i) a left \(R\)-module \(U\) is called \(M\)-flat if the functor \(-\otimes_RU\) preserves the exactness of all short exact sequences with middle term \(M\) and (ii) a right \(R\)-module \(A\) is called \(M\)-injective if the functor \(\Hom_R(-,A)\) preserves the exactness of all short exact sequences with middle term \(M\).NEWLINENEWLINENEWLINEThe author defines a ring \(R\) to be left \(M\)-regular if each left \(R\)-module is \(M\)-flat and gives several characterizations of left \(M\)-regular rings. (Part of his Theorem 1.6 claims that \(R\) is \(M\)-regular precisely when every left cyclic \(R\)-module is \(M\)-flat, but, unfortunately, his argument implies that any module over any ring is a direct sum of cyclics.) He further defines a ring to be a left \(M\)-IF ring if every left injective \(R\)-module is \(M\)-flat and also gives several characterizations of these rings. (For example, \(R\) is left \(M\)-IF precisely when the dual \(A^*\) is \(M\)-injective for every left injective \(R\)-module \(A\).) Characterizations of von Neumann regular rings and IF (injectives are flat) rings are consequences.