A variation of stable range one

Abstract

For a subset S of a ring R, an element a is said to have left S-element stable range 1 if whenever Ra + Rb = R there exists r ∈ S such that a + rb is a unit. When S = R (.resp., S = U ( R ) and S = idem ( R ) )., this is the notion of left stable range 1 (.resp., left unit-1 stable range and left idempotent stable range 1). This article is concerned with S = nil ( R ) .