Stable Range Conditions

Abstract

In this paper, we discuss stable range conditions (stable range one, unit 1-stable range and weakly unit 1-stable range) for a semilocal ring. In particular, we prove that a semilocal ring satisfies weakly unit 1-stable range if and only if it is Σ-clean ring. We also show that none of the properties are local properties.

Key words and phrases. Stable range one,Weakly unit 1-stable range, Unit 1-stable range, Almost unit 1-stable range.

2010 Mathematics Subject Classification. 13A99, 16U99