Covering rings by proper ideals

Abstract: Let $R$ be an associative (not necessarily commutative or unital) ring. A cover by left ideals of $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, we define the left ideal covering number $\eta_\ell(R)$ to be the cardinality of a minimal cover, and we also define $R$ to be $\eta_\ell$-elementary if $\eta_\ell(R) < \eta_\ell(R/I)$ for every nonzero two-sided ideal $I$ of $R$. Similarly, we can define analogous notions for covers by right ideals $\eta_r(R)$ and two-sided ideals $\eta(R)$. In this paper, we classify all $\eta_\ell$-, $\eta_r$-, and $\eta$-elementary rings, and determine their corresponding ideal covering numbers. Since any ring $R$ that admits a finite cover by left ideals has an $\eta_\ell$-elementary residue ring $R/I$ such that $\eta_\ell(R) = \eta_\ell(R/I)$, this completely characterizes such rings and their ideal covering numbers, and likewise for rings coverable by right or two-sided ideals.