Commuting idempotents, square-free modules, and the exchange property

We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A "half-commuting" analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) is an internal exchange ring and idempotents lift modulo J(R), then R is an internal exchange ring. We also clarify some interesting results in the literature by investigating, and ultimately characterizing, the relationships between the finite (internal) exchange property, the (C-3) property, and generalizations of square-free modules. We provide multiple examples delimiting these connections. (C) 2015 Elsevier Inc. All rights reserved.