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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two questions on scalar-reflexive rings


Author: Nicole Snashall
Journal: Proc. Amer. Math. Soc. 116 (1992), 921-927
MSC: Primary 13C13; Secondary 13C05, 13G05
MathSciNet review: 1100664
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Abstract: A module $ M$ over a commutative ring $ R$ with unity is reflexive if the only $ R$-endomorphisms of $ M$ leaving invariant every submodule of $ M$ are the scalar multiplications by elements of $ R$. A commutative ring $ R$ is scalar-reflexive if every finitely generated $ R$-module is reflexive. A local version of scalar-reflexivity is introduced, and it is shown that every locally scalar-reflexive ring is scalar-reflexive. An example is given of a scalar-reflexive domain that is not $ h$-local. This answers a question posed by Hadwin and Kerr. Theorem 7 gives eight equivalent conditions on an $ h$-local domain for it to be scalar-reflexive, thus classifying the scalar-reflexive $ h$-local domains.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1100664-9
PII: S 0002-9939(1992)1100664-9
Article copyright: © Copyright 1992 American Mathematical Society