Contributions to the Taxonomy of Rings
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Ohio University / OhioLINK
2021
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ndltd-OhioLink-oai-etd.ohiolink.edu-ohiou16149955430638652021-08-03T07:16:50Z Contributions to the Taxonomy of Rings Chimal-Dzul, Henry Mathematics Taxonomy of rings Morita context rings prime semiprime reflexive semicommutative 2-primal NI Abelian Dedekind-finite Armendariz polynomial semicommutative ring right real McCoy ring faithful bimodule homomorphism Koethe conjecture 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, along with some examples of finite non-reflexive Dedekind-finite rings that appear early in this dissertation, distinguishes the classes of reflexive and Dedekind-finite rings. 2021-05-26 English text Ohio University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1614995543063865 http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1614995543063865 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
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NDLTD |
| language |
English |
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NDLTD |
| topic |
Mathematics Taxonomy of rings Morita context rings prime semiprime reflexive semicommutative 2-primal NI Abelian Dedekind-finite Armendariz polynomial semicommutative ring right real McCoy ring faithful bimodule homomorphism Koethe conjecture |
| spellingShingle |
Mathematics Taxonomy of rings Morita context rings prime semiprime reflexive semicommutative 2-primal NI Abelian Dedekind-finite Armendariz polynomial semicommutative ring right real McCoy ring faithful bimodule homomorphism Koethe conjecture Chimal-Dzul, Henry Contributions to the Taxonomy of Rings |
| author |
Chimal-Dzul, Henry |
| author_facet |
Chimal-Dzul, Henry |
| author_sort |
Chimal-Dzul, Henry |
| title |
Contributions to the Taxonomy of Rings |
| title_short |
Contributions to the Taxonomy of Rings |
| title_full |
Contributions to the Taxonomy of Rings |
| title_fullStr |
Contributions to the Taxonomy of Rings |
| title_full_unstemmed |
Contributions to the Taxonomy of Rings |
| title_sort |
contributions to the taxonomy of rings |
| publisher |
Ohio University / OhioLINK |
| publishDate |
2021 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1614995543063865 |
| work_keys_str_mv |
AT chimaldzulhenry contributionstothetaxonomyofrings |
| _version_ |
1719458080866959360 |