Centralizer near-rings on infinite groups

Abstract

This work is a continuation of recent work done by Maxson and Smith on finite centralizer near-rings. Our attempt to generalize results which are known to hold for finite centralizer near-rings leads us naturally to a condition on stabilizers of elements of a group which we call the finiteness condition. Assuming this condition a number of recent results are extended to a large class of near-rings. In particular, semisimple centralizer near-rings with finiteness condition are completely characterized. Further, simple centralizer near-rings with finiteness condition are shown to be equivalent to complete 2-primitive centralizer near-rings. Radicals of certain centralizer near-rings are characterized, and all isomorphisms of monogenic centralizer near-rings are determined. Finally, the previous material is applied to the problem of determining when a centralizer near-ring is, in fact, a ring.