. R A ] 1 1 Ju n 20 04 THE COUNTABLE AND FULL EXCHANGE PROPERTIES PACE

We show that cohopfian modules with finite exchange have countable exchange. In particular, a module whose endomorphism ring is Dedekind-finite and π-regular has the countable exchange property. We also show that a module whose en-domorphism ring is Dedekind-finite and regular has full exchange. Finally, working modulo the Jacobson radical, we prove that any module with the (C 2) property and a semi-π-regular endomorphism ring has full exchange. The exchange property for modules was first studied in 1964 by Crawley and Jónsson [CJ], and is defined as follows. A right k-module M k has the ℵ-exchange property if, whenever A = M ⊕ N = i∈I A i , with |I| ℵ, then there are submodules A ′ i ⊆ A i , with A = M ⊕ i∈I A ′ i. If M has ℵ-exchange for all cardinals, then we say it has full exchange. If the same holds just for the finite cardinals, we say that M has finite exchange. It is easy to show that 2-exchange is equivalent to finite exchange. An outstanding problem in module theory is to decide whether or not finite exchange further implies full exchange. It turns out that the finite exchange property is an endomorphism ring invariant; putting E = End(M k), then M k has finite exchange iff E E has finite exchange. A ring, R, such that R R has finite exchange is called an exchange ring, following [Wa], and this turns out to be a left-right symmetric condition. Nicholson [N] calls a ring suitable if, given an equation x + y = 1 there are orthogonal idempotents e ∈ Rx and f ∈ Ry with e+ f = 1. This turns out to be equivalent to R being an exchange ring. It is easy to show that (von Neumann) regular rings, and even π-regular rings, are suitable. Any corner ring in a suitable ring is suitable, and the direct product of suitable rings is suitable. Continuous modules, and hence (quasi-)injective modules, always claim the exchange property [MM 2 ]. Further quasi-continuous modules with finite exchange have full exchange [OR], [MM 3 ]. There are many other classes of modules for which finite exchange implies full exchange, including modules which are direct sums of indecomposibles [ZZ], and modules with abelian endomorphism rings [Ni]. It also turns out that square-free modules (i.e. modules with no submodules isomorphic to a square X ⊕ …