Some properties of finite rings

TL;DR

This paper explores properties of finite rings, proving that if the group of invertible elements is trivial, then the ring must be boolean and commutative, extending classical results about finite division rings.

Contribution

It establishes that finite rings with a trivial group of invertible elements are necessarily boolean and commutative, providing a new characterization of such rings.

Findings

Finite rings with trivial invertible group are boolean.

Such rings are necessarily commutative.

Extends classical results on finite division 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.