Abstract
A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if for every nonzero two-sided ideal I of R. In this paper, we provide the first examples of σ-elementary rings R that have nontrivial Jacobson radical J with R/J noncommutative, and we determine the covering numbers of these rings.
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