Units of integral semigroup rings

Wang, Duzhong (1995) Units of integral semigroup rings. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

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 Artinian 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 bicyclic units generate a subgroup of finite index in U(ZS), for a large class of integral semigroup rings. Because the proof ultimately relies on 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 Higman's result to the case of semigroup rings.

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