CHARACTERIZING ALMOST PERFECT RINGS BY COVERS AND ENVELOPES

Abstract

. Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni–Salce [7] and Bazzoni [4], are gener- alized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs–Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost per- fect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair ( P 1 , D ) (Theorem 4.1). Similar characterization is proved concern- ing the existence of divisible envelopes for h -local rings in the same class (Theorem 5.3). In addition, Bazzoni’s characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.