Unit Groups of a Finite Extension of Galois Rings

Abstract

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.