A ring R is called almost strongly regular if for each a ∈ R, either a or 1 − a is strongly regular. The class of almost strongly regular rings lies properly between the class of strongly regular rings and the class of strongly clean rings. Basic properties and the structure of almost strongly regular rings are studied. We discuss this property for some extensions of rings, including (formal triangular) matrix rings and group rings.
Cui et al. (Thu,) studied this question.