Finite unitary ring with minimal non-nilpotent group of units

Abstract

Let $R$ be a finite unitary ring such that $R=R_0[R^{\ast }]$ where $R_0$ is the prime ring and $R^{\ast }$ is not a nilpotent group. We show that if all proper subgroups of $R^{\ast }$ are nilpotent groups, then the cardinal of $R$ is a power of prime number 2. In addition, if $(R/Jac(R){)}^{\ast }$ 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 $GF(2)$.

10 pages