Zaks' conjecture on rings with semi-regular proper homomorphic images

Abstract

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings [23, Theorem] and Matlis' related result on Prüfer domains [26, Theorem]. It also globalizes Couchot's related result on chained rings [10, Theorem 11]. New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions.