Matrix Algebras over Strongly Non-Singular Rings

TL;DR

This paper investigates rings where torsion-free and non-singular modules coincide, characterizing when matrix rings over such rings are Baer, using properties like semi-heredity and strong non-singularity.

Contribution

It provides a characterization of rings whose matrix rings are Baer, based on the equivalence of torsion-free and non-singular modules and properties like semi-heredity.

Findings

Identifies conditions under which matrix rings are Baer.

Establishes the equivalence of torsion-free and non-singular modules for certain rings.

Connects properties like semi-heredity and strong non-singularity to Baer ring characterization.

Abstract

We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential right ideal $I$ of $R$, and a right $R$-module $M$ is torsion-free if $Tor_{1}^{R}(M,R / Rr)=0$ for every $r \in R$. In particular, we consider a ring $R$ for which the classes of torsion-free and non-singular right $S$-modules coincide for every ring $S$ Morita-equivalent to $R$. We make use of these results, as well as the existence of a Morita-equivalence between a ring $R$ and the $n \times n$ matrix ring $Mat_{n}(R)$, to characterize rings whose $n \times n$ matrix ring is a Baer-ring. A ring is Baer if every right (or left) annihilator is generated by an idempotent. Semi-hereditary, strongly non-singular, and Utumi rings will play an important role, and we explore these concepts and relevant results as well.