A characterization of exchange rings

A necessary and sufficient condition on the endomorphism ring of a module for the module to have the finite exchange property is given. This condition is shown to be strictly weaker than a sufficient condition given by Warfield. The class of rings having these properties is equationally definable and is a natural generalization of the class of regular rings. Finally, it is observed that in the commutative case the category of such rings is equivalent with the category of ringed spaces (X, M) with X a Boolean space and M a sheaf of commutative (not necessarily Noetherian) local rings. In all that follows we will consider modules and morphisms as being over a ring S. Given two direct decompositions K=MeX= ?3 _ {A i E I} of a module, we say that they can be exchanged at M if there are submodules Ac Ai (i e I) such that K=ME() {Aji eI}). A module M is said to have the n-exchange property if any pair of decompositions K=M'eX= ? :,{A i E I} with M'_M and card(I)_n can be exchanged at M'. The module M has the (finite) exchange property if it has the n-exchange property for all (finite) cardinals n. In a fundamental paper [1], Crawley and Jonsson define the exchange properties and use them to prove theorems on isomorphic refinements of direct decompositions of modules. Warfield has shown [3] that an indecomposable module has the exchange property if and only if its endomorphism ring is local. In an attempt to generalize this theorem he has shown recently [4] that if R is Ends(M) and J is its Jacobson radical, then M has the finite exchange property if every principal left ideal of R/J is of the form Re+J where e is an idempotent element in R. In the present paper we will give another condition on R which is, in fact, necessary and sufficient for M to have the finite exchange property and furthermore is strictly weaker than Warfield's. We will frequently use the fact that a direct decomposition K=AEB of a module is naturally related to an idempotent endomorphism 7r of K which leaves the elements of A fixed and has B as its kernel. This will be Presented to the Society, June 28, 1971; received by the editors January 24, 1972. AMS 1970 subject classqicfations. Primary 16A30, 16A32.