Osaka Journal of Mathematics

Gorenstein flat preenvelopes

Alina Iacob

Full-text: Open access

Abstract

We consider a two sided noetherian ring $R$ such that the character modules of Gorenstein injective left $R$-modules are Gorenstein flat right $R$-modules. We then prove that the class of Gorenstein flat right $R$-modules is preenveloping. We also show that the class of Gorenstein flat complexes of right $R$-modules is preenevloping in $\mathit{Ch}(R)$. In the second part of the paper we give examples of rings with the property that the character modules of Gorenstein injective modules are Gorenstein flat. We prove that any two sided noetherian ring $R$ with $\mathop{\mathit{i.d.}}_{R^{\mathrm{op}}} R < \infty$ has the desired property. We also prove that if $R$ is a two sided noetherian ring with a dualizing bimodule ${}_{R}V_{R}$ and such that $R$ is left $n$-perfect for some positive integer $n$, then the character modules of Gorenstein injective modules are Gorenstein flat.

Article information

Source
Osaka J. Math., Volume 52, Number 4 (2015), 895-905.

Dates
First available in Project Euclid: 18 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1447856024

Mathematical Reviews number (MathSciNet)
MR3426620

Zentralblatt MATH identifier
1375.18070

Subjects
Primary: 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25] 18G25: Relative homological algebra, projective classes 18G35: Chain complexes [See also 18E30, 55U15]

Citation

Iacob, Alina. Gorenstein flat preenvelopes. Osaka J. Math. 52 (2015), no. 4, 895--905. https://projecteuclid.org/euclid.ojm/1447856024


Export citation

References

  • D. Bennis and N. Mahdou: Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra 210 (2007), 437–445.
  • E.E. Enochs and O.M.G. Jenda: Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.
  • E.E. Enochs and J.A. López-Ramos: Kaplansky classes, Rend. Sem. Mat. Univ. Padova 107 (2002), 67–79.
  • E.E. Enochs, O.M.G. Jenda and J.A. López-Ramos: Dualizing modules and $n$-perfect rings, Proc. Edinb. Math. Soc. (2) 48 (2005), 75–90.
  • E.E. Enochs, O.M.G. Jenda and J.A. López-Ramos: A noncommutative generalization of Auslander's last theorem, Int. J. Math. Math. Sci. (2005), 1473–1480.
  • E.E. Enochs, S. Estrada and A. Iacob: Gorenstein projective and flat complexes over Noetherian rings, Math. Nachr. 285 (2012), 834–851.
  • E.E. Enochs and A. Iacob: Gorenstein injective covers and envelopes over Noetherian rings, Proc. Amer. Math. Soc. 143 (2015), 5–12.
  • E.E. Enochs and O.M.G. Jenda: Relative Homological Algebra, de Gruyter Expositions in Mathematics 30, de Gruyter, Berlin, 2000.
  • J.R. Garcí a Rozas: Covers and envelopes in the category of complexes of modules, Chapman & Hall/CRC Research Notes in Mathematics 407, Chapman & Hall/CRC, Boca Raton, FL, 1999.
  • H. Holm: Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.
  • H. Holm and P. Jørgensen: Cotorsion pairs induced by duality pairs, J. Commut. Algebra 1 (2009), 621–633.
  • A. Iacob: Gorenstein injective envelopes and covers over two sided noetherian rings, submitted.
  • Y. Iwanaga: On rings with finite self-injective dimension, II, Tsukuba J. Math. 4 (1980), 107–113.