A Characterization of Certain Morphic Trivial Extensions

TL;DR

This paper characterizes when trivial extensions of rings by bimodules are morphic, providing complete descriptions for left perfect rings and commutative reduced rings, and explores their relation to unit regular rings.

Contribution

It offers a complete characterization of morphic trivial extensions for left perfect and commutative reduced rings, extending known results on morphic and unit regular rings.

Findings

Complete characterization for left perfect rings.

Trivial extension by Q/R is morphic for commutative reduced rings.

Extended connections between morphic rings and unit regular rings.

Abstract

Given a ring $R$, we study the bimodules $M$ for which the trivial extension $R\propto M$ is morphic. We obtain a complete characterization in the case where $R$ is left perfect, and we prove that $R\propto Q/R$ is morphic when $R$ is a commutative reduced ring with classical ring of quotients $Q$. We also extend some known results concerning the connection between morphic rings and unit regular rings.