Almost perfect commutative rings

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings R/Rr are perfect rings (in the sense of Bass) for all non-zero-divisorsr∈R. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen-Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.