On rings whose finitely generated left ideals are left annihilators of an element

TL;DR

This paper characterizes various classes of rings based on properties of their finitely generated left ideals, clarifies their relationships, corrects previous results, and fully describes when trivial extensions of commutative domains are morphic.

Contribution

It provides a comprehensive characterization of left pseudo-morphic rings, clarifies their relation to other ring classes, corrects earlier inaccuracies, and determines conditions for trivial extensions to be morphic.

Findings

Characterization of left pseudo-morphic rings

Identification of when pseudo-morphic rings are quasi-morphic or morphic

Complete description of trivial extensions of commutative domains as morphic

Abstract

An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and right morphic. We characterize left pseudo-morphic rings; identify the cases a (left) pseudo morphic ring is (left) quasi-morphic, morphic, Quasi-Frobenius, von Neumann regular, etc.; correct two results in a book and a paper; and completely determine when the trivial extension of a commutative domain is morphic which positively answered a question in a paper.