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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of exchange rings


Author: G. S. Monk
Journal: Proc. Amer. Math. Soc. 35 (1972), 349-353
MSC: Primary 16A48
MathSciNet review: 0302695
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0302695-2
Keywords: Exchange property, isomorphic refinement of decompositions, equational classes of rings, representation of commutative rings
Article copyright: © Copyright 1972 American Mathematical Society