2020/08/30 by Swartz, Eric, Werner, Nicholas J. · 1 citation
#05E16 #13M99 #16P10 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2008.13218
A cover of a unital, associative (not necessarily commutative) 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 show that if R has a finite covering number, then the calculation of σ(R) can be reduced to the case where R is a finite ring of characteristic p and the Jacobson radical J of R has nilpotency 2. Our main result is that if R has a finite covering number and R/J is commutative (even if R itself is not), then either σ(R)=σ(R/J), or σ(R)=pd+1 for some d \geqslant 1. As a byproduct, we classify all commutative σ-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring.