Commutative feebly clean rings

Abstract

A ring [Formula: see text] is defined to be feebly clean, if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text], [Formula: see text] are orthogonal idempotents. Feebly clean rings generalize clean rings and are also a proper generalization of weakly clean rings. The family of all semiclean rings properly contains the family of all feebly clean rings. Further properties of feebly clean rings are studied, some of them analogous to those for clean rings. The feebly clean property is investigated for some rings of complex-valued continuous functions. Throughout, all rings are commutative with identity.