DIVISIBLE ENVELOPES, $\mathcal{P}_1$-COVERS AND WEAK-INJECTIVE MODULES

Abstract

We consider the problem of characterizing the commutative domains such that every module admits a divisible envelope or a [Formula: see text]-cover, where [Formula: see text] is the class of modules of projective dimension at most one. We prove that each one of the two conditions is satisfied exactly for the class of almost perfect domains, that is commutative domains such that all of their proper quotients are perfect rings. Moreover, we show that the class of weak-injective modules over a commutative domain is closed under direct sums if and only if the domain is almost perfect.