Rings as the unions of proper subrings

TL;DR

This paper characterizes how rings can be expressed as unions of three proper subrings and determines the minimal number of proper subrings needed to cover simple matrix rings.

Contribution

It extends group union theorems to rings and finds the minimal covering number of proper subrings for matrix rings.

Findings

All possible unions of three proper subrings of a ring are described.

The minimal number of proper subrings covering the simple matrix ring $M_{n}(q)$ is determined.

Abstract

We describe all possible ways how a ring can be expressed as the union of three of its proper subrings. This is an analogue for rings of a 1926 theorem of Scorza about groups. We then determine the minimal number of proper subrings of the simple matrix ring $M_{n}(q)$ whose union is $M_{n}(q)$.