Covering numbers of commutative rings

Abstract

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 ) = p d + 1 for some d ⩾ 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 .