A characterization of exchange rings
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- by G. S. Monk PDF
- Proc. Amer. Math. Soc. 35 (1972), 349-353 Request permission
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.References
- Peter Crawley and Bjarni Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855. MR 169806
- R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056
- R. B. Warfield Jr., A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. Soc. 22 (1969), 460–465. MR 242886, DOI 10.1090/S0002-9939-1969-0242886-2
- R. B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36. MR 332893, DOI 10.1007/BF01419573
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 349-353
- MSC: Primary 16A48
- DOI: https://doi.org/10.1090/S0002-9939-1972-0302695-2
- MathSciNet review: 0302695