A family of examples of generalized perfect rings

TL;DR

This paper constructs a new family of rings that are semiprimitive but not von Neumann regular, showing that all modules over these rings are quotients of their flat covers by small submodules, answering a specific open question.

Contribution

It introduces a novel family of generalized perfect rings that are semiprimitive and non von Neumann regular, expanding understanding of module theory over such rings.

Findings

All modules are quotients of flat covers by small submodules.

Provides counterexamples to a question by Amini et al. (2007).

Demonstrates existence of non von Neumann regular rings with specific module properties.

Abstract

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A.~Amini, B.~Amini, M.~Ershad and H.~Sharif (2007).