Rings as unions of proper ideals

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.