Rings as unions of proper ideals

TL;DR

This paper introduces and studies the concept of ideal covering numbers in associative rings, constructing examples and characterizing rings with minimal coverings, thereby advancing understanding of ring ideal structures.

Contribution

It defines ideal covering numbers for rings, constructs infinite families of rings achieving minimal bounds, and characterizes rings with the smallest possible ideal coverings.

Findings

Constructed rings with ideal covering number p+1 for each prime p

Characterized all rings with ideal covering number three

Provided insights and open questions on ring ideal coverings

Abstract

In this note, we define and investigate ideal covering numbers of associative rings (not assumed to be commutative or unital): three invariants defined as the minimal number of proper left, right, or two-sided ideals whose union equals the ring. For every prime $p$, we construct four infinite families of rings without identity that attain the sharp lower bound $p + 1$ for ideal covering numbers, each exhibiting distinct behavior with respect to left, right, and two-sided ideal coverings. As a consequence of a result by Lucchini and Mar\'{o}ti, we also characterize all rings with ideal covering numbers three. Finally, we make several observations and propose open questions related to these invariants and the structure of rings admitting such ideal coverings.