In this article, we introduce the normalizer Nu(R)(X) of a subset X of a ring R (with identity) in the unit group u(R) and consider, in particular, the normalizer of the natural basis ±S of the integral semigroup ring N0S of a finite semigroup S. We investigate properties of this normalizer for the class of semigroup rings of inverse semigroups, which contains, for example, matrix rings, in particular, matrix rings over group rings, and partial group rings. We also construct free groups in the unit group of an integral semigroup ring of a Brandt semigroup using a bicyclic unit.
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