Compact rings having a finite simple group of units

Abstract

For a compact Hausdorff ring, one observes that the group of units is a totally disconnected compact topological group and is a finite simple group if and only if it possesses no nontrivial closed normal subgroups. Three classification theorems for compact rings are now given. First, those compact rings with identity having a finite simple group of units are identified. Second, a classification of all compact rings A with identity for which 2 is a unit in A, G modulo the center of G is a finite simple group and the length of W is less than or equal to 4 (or equivalently, W is a torsion group) is given where G is the group of units in A and W is the subgroup of G generated by { g∈ G: g 2 = 1}. Finally, those compact rings with identity having 2 as a unit and for which W is a nilpotent group are identified.