On rings over which every flat left module is finitely projective (Q1176665)

This paper was inspired by a conjecture of Azumaya's that every flat left \(R\)-module is finitely projective if and only if the ascending chain condition holds for chains of annihilators in \(R\) of the form \(Ann(a_ 1)\subseteq Ann(a_ 1a_ 2)\subseteq Ann(a_ 1a_ 2a_ 3)\subseteq\ldots\) where \(a_ 1,a_ 2,a_ 3,\dots\) is a sequence of elements in \(R\). The necessity is known to be true; the author succeeds in giving an example which shows that the sufficiency is untrue. The concept of \(n\)-projectivity is defined and characterizations are obtained for a ring such that every flat left \(R\)-module is \(n\)-projective. It is interesting to note that two of these characterizations too are in terms of certain terminating chains of ideals (of \(M_ n(R)\) and \(R^{(n)}\) respectively). As a corollary, a characterization of rings over which every flat module is finitely projective, is obtained. Another corollary worth mentioning is that if every flat left \(R\)-module is \(n\)-projective, then the same holds for any subring of \(R\). In particular, every subring of a left perfect or a left Noetherian ring has the property that every flat left module is finitely projective. The paper concludes with a discussion on the existence of a certain epimorphism behaving like a projective cover on a module.