C4-modules with the exchange property

Abstract

We prove that if M is a C4-module (D4-module) with the restricted ascending (descending) chain condition on direct summands, then M satisfies the finite exchange iff M is clean, iff M satisfies the full exchange. This result extends an earlier one by the authors on summand-square-free (summand-dual-square-free) modules. Moreover, as an immediate application, it follows that every exchange ring with the restricted DCC on direct summands is clean. The later result is a non-trivial and simultaneous extension of two fundamental results on exchange rings, the first of which is due to Nicholson and says that Abelian exchange rings are clean, and the second is due to Camillo and Yu and says that exchange rings with no infinite sets of orthogonal idempotents are clean.