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

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##### Date

2017-07-22##### Author

McQuaig, Bradley

##### Type of Degree

PhD Dissertation##### Department

Mathematics and Statistics##### Metadata

Show full item record##### 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.

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