A Characterization of Exchange Rings

Abstract

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 $(X,\mathcal {R})$ with X a Boolean space and $\mathcal {R}$ a sheaf of commutative (not necessarily Noetherian) local rings.