Rings with a few more zero-divisors

Abstract

It is well-known that every finite ring with non-zero-divisors has order not exceeding the square of the order n of its left zero-divisor set. Unital rings whose order is precisely n2 have been described already. Here we discuss finite rings with relatively larger zero-divisor sets, namely those of order greater than n3/2. This is achieved by describing the class of all finite rings with left composition length two at most, and using a theorem relating the left composition length of a finite ring to the size of its left zero-divisor set.