On a Class of finite rings

Abstract

In [7], Corbas determined all finite rings in which the product of any two zerodivisors is zero, and showed that they are of two types, one of characteristic p and the other of characteristic p2. The purpose of this paper is to address the problem of the classification of finite rings such that (i) the set of all zero-divisors form an ideal M; (ii) M~ = (0); and (iii) M2 # (0). Because of (i), these rings are called completely primary and urt: shall call a finite completely primary ring R which satisfies conditions (I), (ii) and (iii), a ring wtth property(T). These rings are of three types, niimely, of characteristic p, p2 and p3. The characteristic p2 case is subtLvided into cases in which p E M', p E ann(M) - M' and p E M - an~ri(M), where ann(M) denotes the two-sided annihilator of M in R