On rings with envelopes and covers regarding to C3, D3 and flat modules

Abstract

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.