Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 2, pp. 353-362 (2009)
Remarks on reflexive modules, covers, and envelopes
Richard BelshoffDepartment of Mathematics, Missouri State University, Springfield, Missouri 65897, e-mail: RBelshoff@MissouriState.edu
Abstract: We present results on reflexive modules over Gorenstein rings which generalize results of Serre and Samuel on reflexive modules over regular local rings. We characterize Gorenstein rings of dimension at most two by the property that the dual module $\Hom_R(M, R)$ has G-dimension zero for every finitely generated $R$-module $M$. In the second section we introduce the notions of a reflexive cover and a reflexive envelope of a module. We show that every finitely generated $R$-module has a reflexive cover if $R$ is a Gorenstein local ring of dimension at most two. Finally we show that every finitely generated $R$-module has a reflexive envelope if $R$ is quasi-normal or if $R$ is locally an integral domain.
Keywords: reflexive module, cover, envelope, G-dimension, Gorenstein ring
Classification (MSC2000): 13C13, 13H10; 13D05, 13C10
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Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.