Characterizations of semiperfect and perfect rings
Adapting the notion of a generalized projective cover introduced by \textit{G. Azymaya} [Ring Theory, Proc. Biennial Ohio State-Denison Conf., May 1992, 28-40 (1993; Zbl 0854.16014)], the author defines a module epimorphism \(f:P\to M\) to be a (generalized) cover for \(M\) if \(\text{Ker}(f)\) (\(\text{Ker}(f)\subseteq\text{Rad}(P))\) is superfluous in \(P\). If here \(P\) is also projective then the (generalized) cover is said to be projective and likewise if \(P\) is locally projective. \((P\) is called locally projective if, given any module epimorphism \(g:A\to B\) and any homomorphism \(f:P\to B\), then for every finitely generated submodule \(P_0\) of \(P\) there is a homomorphism \(h:P\to A\) such that \(f=gh\) on \(P_0\).) The paper has three sections. The first gives some basic properties of the concepts just defined. The second uses (generalized) projective covers to give characterizations of semiperfect modules and various generalizations of these. In the final section he uses (generalized) locally projective covers to characterise left perfect, semiperfect and left quasi-perfect rings. (The concept of a quasi-perfect ring was introduced \textit{V. Camillo} and the author [in Commun. Algebra 19, No. 10, 2841-2850 (1991; Zbl 0742.16009)].) In particular he shows that the following statements are equivalent for a ring \(R\): (1) \(R\) is left perfect (semiperfect, left quasi-perfect, resp.); (2) every (finitely generated, artinian resp.) left \(R\)-module has a locally projective cover; (3) every (finitely generated, artinian resp.) left \(R\)-module has a generalized locally projective cover.
scientific article; zbMATH DE number 911929