Units of integral semigroup rings

Thesis (Ph.D.)--Memorial University of Newfoundland, 1995. Mathematics and Statistics Bibliography: leaves 72-74 In this thesis, we study the unit group U(ZS) of the integral semigroup ring ZS of a finite semigroup S. Throughout, we assume that ZS has an identity, and unless mentioned otherwise, it...

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Main Author: Wang, Duzhong, 1959-
Other Authors: Memorial University of Newfoundland. Dept. of Mathematics and Statistics
Format: Thesis
Language:English
Published: 1995
Subjects:
Online Access:http://collections.mun.ca/cdm/ref/collection/theses3/id/6829
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spelling ftmemorialunivdc:oai:collections.mun.ca:theses3/6829 2023-05-15T17:23:32+02:00 Units of integral semigroup rings Wang, Duzhong, 1959- Memorial University of Newfoundland. Dept. of Mathematics and Statistics 1995 vi, 74 leaves Image/jpeg; Application/pdf http://collections.mun.ca/cdm/ref/collection/theses3/id/6829 eng eng Electronic Theses and Dissertations (7.97 MB) -- http://collections.mun.ca/PDFs/theses/Wang_Duzhong.pdf a1078884 http://collections.mun.ca/cdm/ref/collection/theses3/id/6829 The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Paper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries Semigroup rings Unit groups (Ring theory) Text Electronic thesis or dissertation 1995 ftmemorialunivdc 2015-08-06T19:17:40Z Thesis (Ph.D.)--Memorial University of Newfoundland, 1995. Mathematics and Statistics Bibliography: leaves 72-74 In this thesis, we study the unit group U(ZS) of the integral semigroup ring ZS of a finite semigroup S. Throughout, we assume that ZS has an identity, and unless mentioned otherwise, it is assumed that QS is a semisimple Artiiiian ring.--In the first part, we study large subgroups of U(ZS). First, two types of units are introduced: the Bass cyclic units and the bicyclic units. These are appropriate generalizations of the analogous group ring case. In the main theorem, it is shown that, as in the case of integral group rings, both the Bass cyclic units and hieyclic units generate a subgroup of finite index in U(ZS), for a large class of integral semigroup rings. Because the proof ultimately relies ou the celebrated congruence theorems, one has to exclude some semigroups which have some specific Wedderburn simple components of degree 1 or 2 over Q. -- In chapter 3, we deal with some examples. We consider the class of semigroups S that are monoid extensions of elementary Rees semigroups. An algorithm is given to compute generators for the full unit group U(ZS). The algorithm is then applied to a specific example. -- Finally, we classify the semigroups such that the unit group U(ZS) is either finite or has a free subgroup of finite index. The former extends Iligman's result to tin- case of semigroup rings. Thesis Newfoundland studies University of Newfoundland Memorial University of Newfoundland: Digital Archives Initiative (DAI)
institution Open Polar
collection Memorial University of Newfoundland: Digital Archives Initiative (DAI)
op_collection_id ftmemorialunivdc
language English
topic Semigroup rings
Unit groups (Ring theory)
spellingShingle Semigroup rings
Unit groups (Ring theory)
Wang, Duzhong, 1959-
Units of integral semigroup rings
topic_facet Semigroup rings
Unit groups (Ring theory)
description Thesis (Ph.D.)--Memorial University of Newfoundland, 1995. Mathematics and Statistics Bibliography: leaves 72-74 In this thesis, we study the unit group U(ZS) of the integral semigroup ring ZS of a finite semigroup S. Throughout, we assume that ZS has an identity, and unless mentioned otherwise, it is assumed that QS is a semisimple Artiiiian ring.--In the first part, we study large subgroups of U(ZS). First, two types of units are introduced: the Bass cyclic units and the bicyclic units. These are appropriate generalizations of the analogous group ring case. In the main theorem, it is shown that, as in the case of integral group rings, both the Bass cyclic units and hieyclic units generate a subgroup of finite index in U(ZS), for a large class of integral semigroup rings. Because the proof ultimately relies ou the celebrated congruence theorems, one has to exclude some semigroups which have some specific Wedderburn simple components of degree 1 or 2 over Q. -- In chapter 3, we deal with some examples. We consider the class of semigroups S that are monoid extensions of elementary Rees semigroups. An algorithm is given to compute generators for the full unit group U(ZS). The algorithm is then applied to a specific example. -- Finally, we classify the semigroups such that the unit group U(ZS) is either finite or has a free subgroup of finite index. The former extends Iligman's result to tin- case of semigroup rings.
author2 Memorial University of Newfoundland. Dept. of Mathematics and Statistics
format Thesis
author Wang, Duzhong, 1959-
author_facet Wang, Duzhong, 1959-
author_sort Wang, Duzhong, 1959-
title Units of integral semigroup rings
title_short Units of integral semigroup rings
title_full Units of integral semigroup rings
title_fullStr Units of integral semigroup rings
title_full_unstemmed Units of integral semigroup rings
title_sort units of integral semigroup rings
publishDate 1995
url http://collections.mun.ca/cdm/ref/collection/theses3/id/6829
genre Newfoundland studies
University of Newfoundland
genre_facet Newfoundland studies
University of Newfoundland
op_source Paper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries
op_relation Electronic Theses and Dissertations
(7.97 MB) -- http://collections.mun.ca/PDFs/theses/Wang_Duzhong.pdf
a1078884
http://collections.mun.ca/cdm/ref/collection/theses3/id/6829
op_rights The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission.
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