Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 55-73, 2019


$n$-strongly Gorenstein graded modules

Zenghui Gao, Jie Peng

Received April 3, 2017.   Published online May 21, 2018.

Abstract:  Let $R$ be a graded ring and $n\geq1$ an integer. We introduce and study $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n>m$. Many properties of the $n$-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate the relations between the graded and the ungraded $n$-strongly Gorenstein injective (or flat) modules. In addition, the connections between the $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules are considered.
Keywords:  $n$-strongly Gorenstein gr-injective module; $n$-strongly Gorenstein gr-flat module; $n$-strongly Gorenstein gr-projective module
Classification MSC:  16W50, 18G25, 16E05
DOI:  10.21136/CMJ.2018.0160-17


References:
[1] M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-injective and gr-projective modules. Commun. Algebra 26 (1998), 225-240. DOI 10.1080/00927879808826128 | MR 1600686 | Zbl 0895.16020
[2] M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-flat modules. Commun. Algebra 26 (1998), 3195-3209. DOI 10.1080/00927879808826336 | MR 1641595 | Zbl 0912.16022
[3] M. J. Asensio, J. A. López Ramos, B. Torrecillas: Covers and envelopes over gr-Gorenstein rings. J. Algebra 215 (1999), 437-459. DOI 10.1006/jabr.1998.7722 | MR 1686200 | Zbl 0942.16049
[4] M. J. Asensio, J. A. López Ramos, B. Torrecillas: FP-gr-injective modules and gr-FC-rings. Algebra and Number Theory. Proc. Conf., Fez, Morocco (M. Boulagouaz, ed.). Lecture Notes in Pure and Appl. Math. 208, Marcel Dekker, New York (2000), 1-11. DOI 10.1201/9780203903889.ch1 | MR 1724670 | Zbl 0963.16041
[5] M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-injective modules over graded isolated singularities. Commun. Algebra 28 (2000), 3197-3207. DOI 10.1080/00927870008827019 | MR 1765311 | Zbl 0998.16031
[6] M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein modules over Zariski filtered rings. Commun. Algebra 31 (2003), 4371-4385. DOI 10.1081/AGB-120022797 | MR 1995540 | Zbl 1042.16036
[7] M. Auslander, M. Bridger: Stable Module Theory. Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence (1969). DOI 10.1090/memo/0094 | MR 0269685 | Zbl 0204.36402
[8] D. Bennis, N. Mahdou: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210 (2007), 437-445. DOI 10.1016/j.jpaa.2006.10.010 | MR 2320007 | Zbl 1118.13014
[9] D. Bennis, N. Mahdou: A generalization of strongly Gorenstein projective modules. J. Algebra Appl. 8 (2009), 219-227. DOI 10.1142/S021949880900328X | MR 2514856 | Zbl 1176.16008
[10] L. W. Christensen: Gorenstein Dimensions. Lecture Notes in Mathematics 1747. Springer, Berlin (2000). DOI 10.1007/BFb0103980 | MR 1799866 | Zbl 0965.13010
[11] N. Q. Ding, J. L. Chen: The flat dimensions of injective modules. Manuscr. Math. 78 (1993), 165-177. DOI 10.1007/BF02599307 | MR 1202159 | Zbl 0804.16005
[12] N. Q. Ding, J. L. Chen: Coherent rings with finite self-FP-injective dimension. Commun. Algebra 24 (1996), 2963-2980. DOI 10.1080/00927879608825724 | MR 1396867 | Zbl 0855.16001
[13] E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
[14] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. de De Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[15] E. E. Enochs, O. M. G. Jenda, B. Torrecillas: Gorenstein flat modules. J. Nanjing Univ., Math. Biq. 10 (1993), 1-9. MR 1248299 | Zbl 0794.16001
[16] E. E. Enochs, J. A. López Ramos: Gorenstein Flat Modules. Nova Science Publishers, Huntington (2001). MR 2017116 | Zbl 1157.16300
[17] J. R. García Rozas, J. A. López-Ramos, B. Torrecillas: On the existence of flat covers in $R$-gr. Commun. Algebra 29 (2001), 3341-3349. DOI 10.1081/AGB-100105025 | MR 1849490 | Zbl 0992.16034
[18] M. Hermann, S. Ikeda, U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study. Springer, Berlin (1988). DOI 10.1007/978-3-642-61349-4 | MR 0954831 | Zbl 0649.13011
[19] H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
[20] L. X. Mao: Strongly Gorenstein graded modules. Front. Math. China 12 (2017), 157-176. DOI 10.1007/s11464-016-0595-y | MR 3569672 | Zbl 06823674
[21] C. Năstăsescu: Some constructions over graded rings: Applications. J. Algebra 120 (1989), 119-138. DOI 10.1016/0021-8693(89)90192-0 | MR 0977864 | Zbl 0678.16001
[22] C. Năstăsescu, F. Van Oystaeyen: Graded Ring Theory. North-Holland Mathematical Library 28, North-Holland Publishing Company, Amsterdam (1982). MR 0676974 | Zbl 0494.16001
[23] C. Năstăsescu, F. Van Oystaeyen: Methods of Graded Rings. Lecture Notes in Mathematics 1836, Springer, Berlin (2004). DOI 10.1007/b94904 | MR 2046303 | Zbl 1043.16017
[24] B. Stenström: Rings of Quotients. Die Grundlehren der mathematischen Wissenschaften 217. Springer, Berlin (1975). (In German.) MR 0389953 | Zbl 0296.16001
[25] X. Yang, Z. Liu: Strongly Gorenstein projective, injective and flat modules. J. Algebra 320 (2008), 2659-2674. DOI 10.1016/j.jalgebra.2008.07.006 | MR 2441993 | Zbl 1173.16006
[26] X. Yang, Z. Liu: FP-gr-injective modules. Math. J. Okayama Univ. 53 (2011), 83-100. MR 2778885 | Zbl 1222.16029
[27] G. Q. Zhao, Z. Y. Huang: $n$-strongly Gorenstein projective, injective and flat modules. Commun. Algebra 39 (2011), 3044-3062. DOI 10.1080/00927872.2010.496749 | MR 2834145 | Zbl 1247.16007

Affiliations:   Zenghui Gao, Jie Peng, College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, Sichuan Province, P. R. China, e-mail: gaozenghui@cuit.edu.cn, pengjiecuit@163.com;


 
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