Rings Over Which Flat Covers of Finitely Generated Modules are Projective

Abstract

In Bican et al. (2001 Bican , L. , El Bashir , R. , Enochs , E. ( 2001 ). All modules have flat covers . Bull. London Math. Soc. 33 : 385 – 390 .[Crossref], [Web of Science ®] , [Google Scholar]), it is proved that all modules over an arbitrary ring have flat covers. In this article, we shall study rings over which flat covers of finitely generated modules are projective. We call a ring R right almost-perfect if every flat right R-module is projective relative to R. It turns out that a ring is right almost-perfect if and only if flat covers of finitely generated modules are projective. We shall show that the class of almost-perfect rings is properly between the class of perfect and semiperfect rings. We also outline some new characterizations of perfect rings. For example, we show that a ring R is right perfect if every finitely cogenerated right R-module has a projective cover.