# Finite unitary ring with minimal non-nilpotent group of units

## TL;DR

This paper characterizes finite unitary rings with a minimal non-nilpotent group of units, showing their size is a power of 2 and classifying their structure under certain conditions.

## Contribution

It provides a classification of finite unitary rings with minimal non-nilpotent unit groups, including their size and structural isomorphisms.

## Key findings

- Cardinality of the ring is a power of 2.
- If the unit group modulo the Jacobson radical is not a p-group, the ring is isomorphic to specific matrix rings.
- Proper subgroups of the unit group are all nilpotent.

## Abstract

Let $R$ be a finite unitary ring such that $R=R_0[R^*]$ where $R_0$ is the prime ring and $R^*$ is not a nilpotent group. We show that if all proper subgroups of $R^*$ are nilpotent groups, then the cardinal of $R$ is a power of prime number 2. In addition, if $(R/Jac(R))^*$ is not a $p-$group, then either $R\cong M_2(GF(2))$ or $R\cong M_2(GF(2))\oplus A$ where $M_2(GF(2))$ is the ring of $2\times 2$ matrices over the finite field $GF(2)$ and $A$ is a direct sum of finite field $GF(2)$.