Covering numbers of commutative rings

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Swartz, Eric¹  Werner, Nicholas J.² 

[1] William & Mary, Dept Math, Williamsburg, VA 23187 USA

[2] SUNY Coll Old Westbury, Dept Math Comp & Informat Sci, Old Westbury, NY 11568 USA

关键词: Subring cover;    Covering;   

DOI  :  10.1016/j.jpaa.2020.106622

来源: Elsevier

【 摘 要 】

A cover of a unital, associative (not necessarily commutative) ring Ris a collection of proper subrings of Rwhose set-theoretic union equals R. If such a cover exists, then the covering number sigma(R) of R is the cardinality of a minimal cover, and a ring R is called sigma-elementary if sigma(R) < sigma(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 sigma(R) can be reduced to the case where Ris a finite ring of characteristic pand the Jacobson radical Jof Rhas nilpotency 2. Our main result is that if Rhas a finite covering number and R/Jis commutative (even if R itself is not), then either sigma(R) = sigma(R/J), or sigma(R) = p(d) + 1for some d >= 1. As a byproduct, we classify all commutative s-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free