2022/11/18 by Swartz, Eric, Werner, Nicholas J. · 1 citation
#05E16 #16P10 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2211.10313
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 σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R) < σ(R/I) 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.