Centers of semigroup rings and conjugacy classes (Q1109108)

Let \(\Gamma\) be an appropriately partially ordered multiplicative cancellative semigroup which acts as a semigroup of endomorphisms on a skew field K by \(\phi\) : \(\Gamma\) \(\to End K\). The skew semigroup power series ring K[[\(\Gamma\) ;\(\phi\) ]] consists of all formal sums \(\alpha =\sum s\alpha (s)\), \(s\in \Gamma\), \(\alpha\) (s)\(\in K\) whose supports \(\{\) \(s\in \Gamma |\) \(\alpha (s)=0\}\) are a union of a finite number of chains satisfying the ascending chain condition, and where \(ks=sk^{\phi s}\) for \(k\in K\). Facts about the ordinary skew semigroup ring K[\(\Gamma\) ;\(\phi\) ] are corollaries of more general results by specializing \(\Gamma\) to be discretely or trivially ordered. First, the notion of the conjugate of an element is generalized from a group to a semigroup. This leads to some natural little known subsemigroups of \(\Gamma\). Then later these concepts are used to describe the centers of skew semigroup related rings such that K[\(\Gamma\) ;\(\phi\) ]\(\subseteq K[[\Gamma;\phi]]\).