Two questions on scalar-reflexive rings

Snashall, Nicole

doi:10.1090/S0002-9939-1992-1100664-9

Two questions on scalar-reflexive rings
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by Nicole Snashall

Proc. Amer. Math. Soc. 116 (1992), 921-927

DOI: https://doi.org/10.1090/S0002-9939-1992-1100664-9

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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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