2021/12/03 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.2112.01667
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. If R is a ring with unity, then we define the unital covering number σu(R) to be the size of a minimal cover of R by subrings that contain 1R (if such a cover exists), and R is σu-elementary if σu(R) < σu(R/I) for every nonzero two-sided ideal of R. In this paper, we classify all σ-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σu-elementary rings and prove σu(R) = σ(R) for every σu-elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R' such that σ(R) = σu(R'), which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if \mathscrE(N) := \m : m ≤ N, σ(R) = m for some ring R\, then we show that |\mathscrE(N)| = Θ(N/log(N)), which proves that almost all integers are not covering numbers of a ring.