Osaka Journal of Mathematics

Torsionfree dimension of modules and self-injective dimension of rings

Chonghui Huang and Zhaoyong Huang

Full-text: Open access

Abstract

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n \geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_{R}^{i}(M, R)=0$ for any $1 \leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$.

Article information

Source
Osaka J. Math., Volume 49, Number 1 (2012), 21-35.

Dates
First available in Project Euclid: 21 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1332337236

Mathematical Reviews number (MathSciNet)
MR2903252

Zentralblatt MATH identifier
1244.16007

Subjects
Primary: 16E10: Homological dimension 16E05: Syzygies, resolutions, complexes

Citation

Huang, Chonghui; Huang, Zhaoyong. Torsionfree dimension of modules and self-injective dimension of rings. Osaka J. Math. 49 (2012), no. 1, 21--35. https://projecteuclid.org/euclid.ojm/1332337236


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