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

TL;DR

This paper extends Zaks' conjecture from integral domains to arbitrary rings, providing new insights and examples of rings with semi-regular proper homomorphic images, and unifying several related results.

Contribution

It generalizes Zaks' conjecture to all rings, broadening the scope of previous results on semi-regular homomorphic images.

Findings

Extended Zaks' conjecture to arbitrary rings.

Unified results on Noetherian, Prufer, and chained rings.

Constructed new examples via trivial ring extensions.

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 and Matlis' related result on Prufer domains. It also globalizes Couchot's related result on chained rings. New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions.