$k$-torsionless modules with finite Gorenstein dimension

Title:

$k$-torsionless modules with finite Gorenstein dimension

Creator:

Salimi, Maryam, Elham, and Yassemi, Siamak

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:dcb51189-3d91-4b76-8cbd-afb0b3a00878
uuid:dcb51189-3d91-4b76-8cbd-afb0b3a00878

Subject:

torsionless module, reflexive module, and Gorenstein dimension

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{\mathfrak p}$ is reflexive for ${\mathfrak p} \in {\rm Spec}(R) $ with ${\rm depth}(R_{\mathfrak p}) \leq 1$, and ${\mbox {G-{\rm dim}}}_{R_{\mathfrak p}} (M_{\mathfrak p}) \leq {\rm depth}(R_{\mathfrak p})-2 $ for ${\mathfrak p}\in {\rm Spec} (R) $ with ${\rm depth}(R_{\mathfrak p})\geq 2 $. This gives a generalization of Serre and Samuel's results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\geq 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every $R$-module has a $k$-torsionless cover provided $R$ is a $k$-Gorenstein ring.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

663-672

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:3

Harvested from:

CDK

Metadata only:

false