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, #) with X a Boolean space and M a sheaf of commutative (not necessarily Noetherian) local rings.