Morita-Equivalence Between Strongly Non-Singular Rings and the Structure of the Maximal Ring of Quotients

Abstract

This dissertation focuses on extending certain notions from Abelian group theory and module theory over integral domains to modules over non-commutative rings. In particular, we investigate generalizations of torsion-freeness and characterize rings for which torsion-freeness and non-singularity coincide under a Morita-equivalence. 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 $\Tor_{1}^{R}(M,R \slash Rr)=0$ for every $r \in R$. Incidentally, we find that this is related to characterizing rings for which the $n \times n$ matrix ring $Mat_{n}(R)$ is a Baer-ring. A ring is Baer if every right (or left) annihilator is generated by an idempotent. Strongly non-singular and semi-hereditary rings play a vital role, and we consider relevant examples and related results.

This leads to a discussion of divisible modules and two-sided submodules of the maximal ring of quotients Q. As with torsion-freeness, there are various notions of divisibility in the general setting, and we consider rings for which these various notions coincide. More specifically, we consider the structure of Q/R in the case that its projective dimension is at most 1, and R is a right and left duo domain. A ring R is a right (left) duo ring if $Ra \subseteq aR \ (aR \subseteq Ra)$ for every $a \in R$. In this setting, we find that h-divisibility and classical divisibility coincide, and Q/R can be decomposed into a direct sum of countably-generated two-sided R-submodules. We consider related results, as well as examples of such rings.