  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.
