Commuting idempotents, square-free modules, and the exchange property. (Q745147)

The paper has three main results. Firstly, a criterion is obtained for when idempotents of a ring \(R\) which commute modulo the Jacobson radical \(J(R)\) can be lifted to commuting idempotents of \(R\). Secondly, it is proved that if \(R/J(R)\) is an internal exchange ring and idempotents lift modulo \(J(R)\), then \(R\) is an internal exchange ring. Lastly, the authors establish and clarify inclusion relations between square-free modules, modules with (finite) exchange property, modules with \(C_3\) property, and modules that possess other related properties.