Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-11T11:53:51.114Z Has data issue: false hasContentIssue false
  • English
  • Français

Homological Properties Relative to Injectively Resolving Subcategories

Published online by Cambridge University Press:  20 November 2018

Zenghui Gao
Zenghui Gao*
Affiliation:
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P.R. China e-mail: gaozenghui@cuit.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\text{Ext}_{R}^{1}\left( G,\,M \right)\,=\,0$ (resp. $\text{Tor}_{1}^{R}\left( N,\,G \right)\,=\,0$) for any $G\,\in \,\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that ${{\mathcal{E}}^{+}}\,\subseteq \,\mathcal{F}$ and ${{\mathcal{F}}^{+}}\,\subseteq \,\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

Keywords

16E3016E1016E60injectively resolving subcategoryε-injective module (dimension)ε-flat module (dimension)coverpreenvelopeε-(semi)hereditary ring
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 741 - 756
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Bican, L., El Bashir, R. and Enochs, E. E., All modules have flat covers. Bull. London Math. Soc. 33(2001), 385–390. http://dx.doi.Org/10.1017/S0024609301008104 Google Scholar
[2] Christensen, L. W., Gorenstein dimensions. Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.Google Scholar
[3] Colby, R. R., Rings which have flat injective modules. J. Algebra 35(1975), 239–252. http://dx.doi.Org/10.101 6/0021 -8693(75)90049-6 Google Scholar
[4] Ding, N. Q. and Chen, J. L., The flat dimensions of injective modules. Manuscripta Math. 78(1993), 165–177. http://dx.doi.Org/10.1007/BF02599307 Google Scholar
[5] Eklof, P. C. and Trlifaj, J., Covers induced by Ext1. J. Algebra 231(2000), 640–651. http://dx.doi.Org/10.1006/jabr.2000.8343 Google Scholar
[6] Emmanouil, I., On thefiniteness of Gorenstein homological dimensions. J. Algebra 372(2012), 376–396. http://dx.doi.Org/10.1016/j.jalgebra.2O12.09.018 Google Scholar
[7] Enochs, E. E., Injective and flat covers, envelopes and resolvents. Israel J. Math. 39(1981), 189–209. http://dx.doi.Org/10.1007/BF02760849 Google Scholar
[8] Enochs, E. E. and Jenda, O. M. G. Copure injective modules. Quaestiones Math. 14(1991), 401–409. http://dx.doi.Org/10.1080/16073606.1991.9631658 Google Scholar
[9] Enochs, E. E. and Jenda, O. M. G., Copure injective resolutions, flat resolvents and dimensions. Comment. Math. Univ. Carolin. 34(1993), 203–211.Google Scholar
[10] Enochs, E. E. and Jenda, O. M. G., Gorenstein injective andprojective modules. Math. Z. 220(1995), 611–633. http://dx.doi.Org/10.1007/BF02572634 Google Scholar
[11] Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000.Google Scholar
[12] Faith, C., Algebra I: Rings, modules and categories. Springer-Verlag, Berlin-New York, 1973.Google Scholar
[13] Gao, Z. H., On GI-injective modules. Comm. Algebra 40(2012), 3841–3858. http://dx.doi.Org/10.1080/0092 7872.2011.597809 Google Scholar
[14] Gao, Z. H., On GI-flat modules and dimensions. J. Korean Math. Soc. 50(2013), 203–218. http://dx.doi.Org/10.4134/JKMS.2013.50.1.203 Google Scholar
[15] Gao, Z. H. and Huang, Z. Y., Weak injective covers and dimensions of modules. Acta Math. Hungar., to appear.Google Scholar
[16] Gao, Z. H. and Wang, F. G., Weak injective and weak flat modules. Comm. Algebra 43(2015), 3857–3868. http://dx.doi.Org/10.1080/00927872.2014.92412 8 Google Scholar
[17] Holm, H., Gorenstein homological dimensions. J. Pure Appl. Algebra 189(2004), 167–193. http://dx.doi.Org/10.1016/j.jpaa.2003.11.007 Google Scholar
[18] Hu, K. and Wang, F. G., Some results on Gorenstein Dedekind domains and their factor rings. Comm. Algebra 41(2013), 284–293. http://dx.doi.Org/10.1080/00927872.2011.629268 Google Scholar
[19] Huang, Z. Y., Homological dimensions relative to preresolving subcategories, Kyoto J. Math. 54(2014), 727–757. http://dx.doi.Org/10.1215/21562261-2801795 Google Scholar
[20] Khashyarmanesh, K. and Salarian, Sh., On the rings whose injective hulls are flat. Proc. Amer. Math. Soc. 131(2003), 2329–2335. http://dx.doi.Org/10.1090/S0002-9939-03-06829-1 Google Scholar
[21] Mao, L. and Ding, N. Q., The cotorsion dimension of modules and rings. In: Abelian groups, rings, modules, and homological algebra, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 217–233.Google Scholar
[22] Mao, L. and Ding, N. Q., FI-injective and FI-flat modules. J. Algebra 309(2007), 367–385. http://dx.doi.Org/10.1016/j.jalgebra.2006.10.019 Google Scholar
[23] Mao, L. and Ding, N. Q., On divisible and torsionfree modules. Comm. Algebra 36(2008), 708–731. http://dx.doi.Org/10.1080/00927870701724201 Google Scholar
[24] Mahdou, N., and Tamekkante, M., On (strongly) Gorenstein (semi)hereditary rings. Arab. J. Sci. Eng. 36(2011), 431–440. http://dx.doi.Org/10.1007/s13369-011-0047-7 Google Scholar
[25] Rotman, J. J., An introduction to homological algebra. Pure and Applied Mathematics, Academic Press, New York, 1979.Google Scholar
[26] Stenstrôm, B., Coherent rings and FP-injective modules. J. London Math. Soc. 2(1970), 323–329. http://dx.doi.Org/10.1112/jlms/s2-2.2 323 Google Scholar
[27] Xu, J. Z., Flat covers of modules. Lecture Notes in Mathematics, 1634, Springer-Verlag, Berlin, 1996.Google Scholar