Some properties of finite rings

Abstract

A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as possible. Here we will be particularly interested in the case where this group is as small as possible, namely reduced to 1. We will show that, if this is the case, then the ring is boolean. Thus, here too, the ring is commutative.