Matrix Algebras over Strongly Non-Singular Rings

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 $To{r}_1^R(M,R\mathrm{/}Rr)=0$ 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 $Ma{t}_n(R)$, 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.