Automorphisms of a certain class of completely primary finite rings (Q2895913)

A completely primary finite ring is a ring \(R\) with identity \(1 \neq 0\) whose subset of all zero-divisors forms a unique maximal ideal \(J\). Let \(R\) be a completely primary finite ring with maximal ideal \(J\). Then \(R\) is of order \(p^{nr}\); \(J\) is the Jacobson radical of \(R\); \(J^m = (0)\), where \(m \leq n\), and the residue field \(R/J\) is a finite field \(GF(p^r)\), for some prime \(p\) and positive integers \(n, r\). Let \(R'\) be the Galois ring of the form \(GR(p^{nr}, p^n)\). For each \(i = 1, \dots, h\) , let \(u_i \in J(R)\), such that \(U\) is an \(h\)-dimensional \(R'\)-module generated by \(\{u_1, \dots, u_h\}\) so that \(R = R' \oplus U\) is an additive group. A completely primary finite ring \(R\) which satisfies the three properties (i) \(J(R) = pR' \oplus U\), (ii) \((J(R))^{n-1} = p^{n-1}R'\), (iii) \((J(R))^n = (0)\), is called a ring with property A. In this paper the author determined the automorphisms of completely primary finite rings with property A.