## Automorphisms of completely primary finite rings of characteristic $p$

### Volume 111 / 2008

Colloquium Mathematicum 111 (2008), 91-113
MSC: Primary
16P10, 20B25; Secondary 16L99, 12E20.
DOI: 10.4064/cm111-1-9

#### Abstract

A completely primary ring is a ring $R$ with identity $1\neq 0$ whose subset of zero-divisors forms the unique maximal ideal ${\cal J}$. We determine the structure of the group of automorphisms ${\rm Aut}(R)$ of a completely primary finite ring $R$ of characteristic $p,$ such that if ${\cal J}$ is the Jacobson radical of $R,$ then ${\cal J}^{3}=(0),$ ${\cal J}^{2}\neq (0),$ the annihilator of ${\cal J}$ coincides with ${\cal J}^{2}$ and $R/{\cal J}\cong {\rm GF}(p^{r}),$ the finite field of $p^{r}$ elements, for any prime $p$ and any positive integer $r.$