On the weak regularity of semigroup rings
A ring \(R\) is said to be weakly regular iff, for every \(a \in R\), there exists \(x \in R^ 1 aR^ 1\) such that \(a = xa\). The aim of the paper is to investigate weakly regular semigroup rings. So far there is only one known result in this direction: Vanaja (1970) and Fisher (1974) described weakly regular group rings (this description may be obtained from the well known description of regular group rings by changing ``regular'' for ``weakly regular''). The author obtains many results giving sufficient conditions for some semigroup rings to be weakly regular. The full description of weakly regular semigroup rings \(R[S]\) is given only for a locally finite inverse semigroup \(S\), and for \(S\) a generalized Brandt semigroup (i.e. an orthodox completely 0-simple semigroup).
scientific article; zbMATH DE number 549059