## Osaka Journal of Mathematics

### Torsionfree dimension of modules and self-injective dimension of rings

#### 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

#### 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

#### References

• M. Auslander and M. Bridger: Stable Module Theory, Memoirs of the American Mathematical Society 94, Amer. Math. Soc., Providence, RI, 1969.
• \begingroup L.W. Christensen, A. Frankild and H. Holm: On Gorenstein projective, injective and flat dimensions–-a functorial description with applications, J. Algebra 302 (2006), 231–279. \endgroup
• R.R. Colby and K.R. Fuller: A note on the Nakayama conjectures, Tsukuba J. Math. 14 (1990), 343–352.
• M. Hoshino: Algebras of finite self-injective dimension, Proc. Amer. Math. Soc. 112 (1991), 619–622.
• M. Hoshino: Reflexive modules and rings with self-injective dimension two, Tsukuba J. Math. 13 (1989), 419–422.
• Z.Y. Huang: Extension closure of $k$-torsionfree modules, Comm. Algebra 27 (1999), 1457–1464.
• \begingroup Z.Y. Huang: Selforthogonal modules with finite injective dimension, Sci. China Ser. A 43 (2000), 1174–1181. \endgroup
• Z.Y. Huang: Approximation presentations of modules and homological conjectures, Comm. Algebra 36 (2008), 546–563.
• \begingroup Z.Y. Huang and O. Iyama: Auslander-type conditions and cotorsion pairs, J. Algebra 318 (2007), 93–100. \endgroup
• Z.Y. Huang and G.H. Tang: Self-orthogonal modules over coherent rings, J. Pure Appl. Algebra 161 (2001), 167–176.
• Y. Iwanaga: On rings with finite self-injective dimension II, Tsukuba J. Math. 4 (1980), 107–113.
• O. Iyama: $\tau$-categories III, Auslander orders and Auslander–Reiten quivers, Algebr. Represent. Theory 8 (2005), 601–619.
• J.P. Jans: On finitely generated modules over Noetherian rings, Trans. Amer. Math. Soc. 106 (1963), 330–340.
• D.A. Jorgensen and L.M. Şega: Independence of the total reflexivity conditions for modules, Algebr. Represent. Theory 9 (2006), 217–226.
• J.J. Rotman: An Introduction to Homological Algebra, Academic Press, New York, 1979.
• A. Zaks: Injective dimension of semi-primary rings, J. Algebra 13 (1969), 73–86.