Covering Numbers of Finite Rings

Abstract

A noncyclic finite group is always equal to a union of its proper subgroups, and the minimum number of subgroups necessary to achieve this union is called the covering number of the group. Here, we investigate the analogous ideas for finite rings. We say an associative ring R with unity is coverable if it is equal to a union of its proper subrings. If this can be done using a finite number of proper subrings, then the covering number of R is the minimum number of subrings required to cover R. Not every ring is coverable, and even when R is finite it is nontrivial to decide whether R is coverable. We present a classification theorem for finite coverable semisimple rings and determine the covering number for R when R is coverable and equal to a direct product of finite fields.