A variation of stable range one

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).