Units in orders and integral semigroup rings

We continue the investigations on the finite conjugacy centre, denoted as , the hyper centre, and thus also the second centre , of the unit group of a -order Γ in a finite-dimensional -algebra. In particular, it is shown that is finitely generated and , also conditions are obtained for these two groups to be equal or central. Applications are given to the unit group of the integral semigroup ring of a finite semigroup. It follows that often has central height two and . This extends earlier results on integral group rings by Arora and Passi, Bovdi, Jespers and Juriaans, and Li and Parmenter.