On flat and projective envelopes
The answer to the question as to over which rings each module has a flat envelope, is not known, but some results on this question have been obtained since 1981. This paper studies this question as well, and though a full answer is not found, the authors prove a characterization of rings over which every right module has a flat envelope which is a monomorphism. They also show that if \(R\) is a right perfect and left coherent ring, then every right \(R\)-module has a flat envelope. As corollary to this result it is found that if \(R\) is a left artinian ring, then every right \(R\)-module has a flat envelope. A final proposition proves that every right \(R\)-module has a projective (pre)envelope if and only if \(R\) is a right perfect and left coherent ring. The paper is concluded by a method to find the projective (pre)envelope of a finitely generated right module over an arbitrary left artinian ring, and an example.
scientific article; zbMATH DE number 554132