Doctor of Philosophy (PhD), Ohio University, 2021, Mathematics (Arts and Sciences)

The present work constitutes some contributions to the taxonomy of various classes of non-commutative rings. These include the classes of reversible, reflexive, semicommutative, 2-primal, NI, abelian, Dedekind finite, Armendariz, and McCoy rings. In addition, two new families of rings are introduced, namely right real McCoy and polynomial semicommutative. The first contribution concerns the hierarchy and interconnections of reflexive, abelian, and semicommutative rings in the setting of finite rings. It is shown that a minimal reflexive abelian non-semicommutative ring has order 256 and that the group algebra F_2D_8 is an example of such a ring. This answers an open question posed by Professor Steve Szabo in his paper on a taxonomy of finite 2-primal rings. The second set of contributions includes characterizations of reflexive, 2-primal, weakly 2-primal and NI rings in the setting of Morita context rings. The results on Morita context rings which are reflexive are shown to generalize known characterizations of prime and semiprime Morita context rings. Similarly, the results on 2-primal, weakly 2-primal and NI Morita context rings presented here generalize various known results for upper triangular matrix rings. Specifically, a characterization of NI Morita context rings is shown to be equivalent to the (most) famous conjecture in Ring Theory: Kothe's conjecture. The third collection of contributions addresses a study of right real McCoy rings and polynomial semicommutative rings; both classes of rings being introduced in this dissertation. The correlations between these types of rings is described, and a series of examples of finite polynomial semicommutative rings is given. In addition, equivalent conditions to the McCoy condition and some of its variations are addressed. Finally, the last contribution establishes that the rings of column finite matrices, and row and column finite matrices over a reflexive ring are reflexive non-Dedekind finite. This result, alon (open full item for complete abstract)

Committee: Sergio López-Permouth Dr. (Advisor); Winfried Just Dr. (Committee Member); Gulisashvilli Archil Dr. (Committee Member); Jeffrey Dill Dr. (Committee Member) Subjects: Mathematics

Doctor of Philosophy (PhD), Ohio University, 2005, Mathematics (Arts and Sciences)

A right R-module M is called CS if every submodule of M is essential in a direct summand of M. In this dissertation, we study certain classes of CS and Σ-CS rings and modules. A ring R is called right (left) max-min CS if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of R is a direct summand of R. Among other results, it is shown that if R is a nondomain prime ring, then R is right nonsingular, right max-min CS with a uniform right ideal if and only if R is a left nonsingular, left max-min CS with a uniform left ideal. This result gives, in particular, Huynh, Jain and Lopez-Permouth Theorem for prime rings of finite uniform dimension. Also we show that a nondomain right nonsingular prime ring with a uniform right ideal is right finitely Σ-min-CS if every finitely generated right ideal of R is min CS. Jain, Kanwar and Lopez-Permouth characterized right nonsingular semiperfect right CS rings. We obtain the structure of right nonsingular semiperfect right min CS rings with a uniform right ideal. It is shown that such rings are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. As a consequence, we show that an indecomposable right nonsingular semiperfect ring is right CS if and only if it is min CS with a uniform right ideal. We generalize this result to endomorphism rings of nonsingular semiperfect progenerator min CS modules with a uniform submodule. It is known that every Σ-CS module is a direct sum of uniform modules and countably Σ-CS modules need not be Σ-CS. A sufficient condition that guarantees a countably Σ-CS module, which is a direct sum of uniform modules, to be Σ-CS has been obtained.

Committee: S. Jain (Advisor) Subjects: Mathematics; Mathematics

Doctor of Philosophy, University of Toledo, 2011, Mathematics

Kashkarev has shown that the mod 2 Steenrod algebra is a prime ring. For any odd prime p, we prove that the mod p Steenrod algebra is also a prime ring. In sequel, for any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension.

Committee: Charles Odenthal PhD (Committee Chair); Paul Hewitt PhD (Committee Member); John Palmieri PhD (Committee Member); Martin Pettet PhD (Committee Member) Subjects: Mathematics