Rocky Mountain Journal of Mathematics

On Ding injective, Ding projective and Ding flat modules and complexes

James Gillespie

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along the way, we generalize some results of Stovicek and Bravo, Gillespie and Hovey to obtain other interesting corollaries. For example, we show that, over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules, that is, $Ext ^1_R(F,Z_nI) = 0$ for any such complex $I$ and flat module $F$. On the other hand, over any coherent ring, the cycles of any exact complex $P$ with projective components must satisfy $Ext ^1_R(Z_nP,A) = 0$ for any absolutely pure module~$A$.

Article information

Rocky Mountain J. Math., Volume 47, Number 8 (2017), 2641-2673.

First available in Project Euclid: 3 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16D80: Other classes of modules and ideals [See also 16G50]
Secondary: 16E05: Syzygies, resolutions, complexes

Ding projective Ding injective


Gillespie, James. On Ding injective, Ding projective and Ding flat modules and complexes. Rocky Mountain J. Math. 47 (2017), no. 8, 2641--2673. doi:10.1216/RMJ-2017-47-8-2641.

Export citation


  • J. Adámek and J. Rosický, Locally presentable and accessible categories, Lond. Math. Soc. Lect. Note 189, Cambridge University Press, Cambridge, 1994.
  • Daniel Bravo and James Gillespie, Absolutely clean, level, and Gorenstein AC-injective complexes, Comm. Algebra 44 (2016), 2213–2233.
  • Daniel Bravo, James Gillespie and Mark Hovey, The stable module category of a general ring, submitted.
  • N. Ding and J. Chen, The flat dimensions of injective modules, Manuscr. Math. 78 (1993), 165–177.
  • N. Ding and J. Chen, Coherent rings with finite self- FP-injective dimension, Comm. Alg. 24 (1996), 2963–2980.
  • N. Ding, Y. Li and L. Mao, Strongly Gorenstein flat modules, J. Australian Math. Soc. 86 (2009), 323–338.
  • N. Ding and L. Mao, Reletive FP-projective modules, Comm. Alg. 33 (2005), 1587–1602.
  • ––––, Envelopes and covers by modules of finite FP-injective and flat dimensions, Comm. Alg. 35 (2007), 833–849.
  • ––––, Gorenstein FP-injective and Gorenstein flat modules, J. Alg. Appl. 7 (2008), 491–506.
  • E. Enochs, S. Estrada and A. Iacob, Gorenstein projective and flat complexes over noetherian rings, Math. Nachr. 285 (2012), 834–851.
  • E. Enochs and J.R. García-Rozas, Tensor products of chain complexes, Math J. Okayama Univ. 39 (1997), 19–42.
  • E. Enochs, A. Iacob, and O.M.G. Jenda, Closure under transfinite extensions, Illinois J. Math. 51 (2007), 561–569.
  • Edgar E. Enochs and Overtoun M.G. Jenda, Relative homological algebra, de Gruyter Expos. Math. 30, Walter de Gruyter & Co., Berlin, 2000.
  • J.R. García-Rozas, Covers and envelopes in the category of complexes of modules, Res. Notes Math. 407, Chapman & Hall/CRC, Boca Raton, FL, 1999.
  • James Gillespie, The flat model structure on Ch$($R$)$, Trans. Amer. Math. Soc. 356 (2004), 3369–3390.
  • ––––, Kaplansky classes and derived categories, Math. Z. 257 (2007), 811–843.
  • ––––, Model structures on modules over Ding-Chen rings, Homol. Homot. Appl. 12 (2010), 61–73.
  • ––––, How to construct a Hovey triple from two cotorsion pairs, Fund. Math. 230 (2015), 281–289.
  • ––––, Gorenstein complexes and recollements from cotorsion pairs, Adv. Math. 291 (2016), 859–911.
  • Rüdiger G öbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, de Gruyter Expos. Math. 41, Walter de Gruyter & Co., Berlin, 2006.
  • Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), 553–592.
  • Y. Iwanaga, On rings with finite self-injective dimension, Comm. Alg. 7 (1979), 393–414.
  • ––––, On rings with finite self-injective dimension II, Tsukuba J. Math. 4 (1980), 107–113.
  • T.Y. Lam, Lectures on modules and rings, Grad. Texts Math. 189, Springer-Verlag, New York, 1999.
  • Bo Stenstr öm, Coherent rings and FP-injective modules, J. Lond. Math. Soc. 2 (1970), 323–329.
  • ––––, Rings of quotients, Grundl. Math. Wissen. Einz. 217, Springer-Verlag, New York, 1975.
  • Jan Š\vtovíček, On purity and applications to coderived and singularity categories, arXiv:1412.1615.
  • Charles A. Weibel, An introduction to homological algebra, Cambr. Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994.
  • Gang Yang and Zhongkui Liu, Cotorsion pairs and model structures on $\ch$, Proc. Edinburgh Math. Soc. 54 (2011), 783–797.
  • Gang Yang, Zhongkui Liu and Li Liang, Ding projective and Ding injective modules, Alg. Colloq. 20 (2013), 601–612.
  • ––––, Model structures on categories of complexes over Ding-Chen rings, Comm. Alg. 41 (2013), 50–69.