Countably coverable rings

Abstract

Let R be an associative ring. Then R is said to be coverable provided R is the union of its proper subrings (which we do not require to be unital even if R is so). One verifies easily that R is coverable if and only if R is not generated as a ring by a single element. In case R can be expressed as the union of a finite number of proper subrings, the least such number is called the covering number of R. Covering numbers of rings have been studied in a series of recent papers. The purpose of this note is to study rings which can be covered by a countable collection of subrings.

KEYWORDS:

• Algebraically closed field
• coverable ring
• localization
• transcendence basis

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A70
• 13E05
• Secondary: 13G05

Acknowledgments

The authors thank the referee for a timely review and for comments which improved the exposition of this paper.