| Title:
|
$n$-strongly Gorenstein graded modules (English) |
| Author:
|
Gao, Zenghui |
| Author:
|
Peng, Jie |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
55-73 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a graded ring and $n\geq 1$ 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. (English) |
| Keyword:
|
$n$-strongly Gorenstein gr-injective module |
| Keyword:
|
$n$-strongly Gorenstein gr-flat module |
| Keyword:
|
$n$-strongly Gorenstein gr-projective module |
| MSC:
|
16E05 |
| MSC:
|
16W50 |
| MSC:
|
18G25 |
| idZBL:
|
Zbl 07088769 |
| idMR:
|
MR3923574 |
| DOI:
|
10.21136/CMJ.2018.0160-17 |
| . |
| Date available:
|
2019-03-08T14:55:38Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147617 |
| . |
| Reference:
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[1] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: Gorenstein gr-injective and gr-projective modules.Commun. Algebra 26 (1998), 225-240. Zbl 0895.16020, MR 1600686, 10.1080/00927879808826128 |
| Reference:
|
[2] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: Gorenstein gr-flat modules.Commun. Algebra 26 (1998), 3195-3209. Zbl 0912.16022, MR 1641595, 10.1080/00927879808826336 |
| Reference:
|
[3] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: Covers and envelopes over gr-Gorenstein rings.J. Algebra 215 (1999), 437-459. Zbl 0942.16049, MR 1686200, 10.1006/jabr.1998.7722 |
| Reference:
|
[4] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: FP-gr-injective modules and gr-FC-rings.Algebra and Number Theory. Proc. Conf., Fez, Morocco M. Boulagouaz Lecture Notes in Pure and Appl. Math. 208, Marcel Dekker, New York (2000), 1-11. Zbl 0963.16041, MR 1724670, 10.1201/9780203903889.ch1 |
| Reference:
|
[5] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: Gorenstein gr-injective modules over graded isolated singularities.Commun. Algebra 28 (2000), 3197-3207. Zbl 0998.16031, MR 1765311, 10.1080/00927870008827019 |
| Reference:
|
[6] Asensio, M. J., Ramos, J. A. López, Torrecillas, B.: Gorenstein modules over Zariski filtered rings.Commun. Algebra 31 (2003), 4371-4385. Zbl 1042.16036, MR 1995540, 10.1081/AGB-120022797 |
| Reference:
|
[7] Auslander, M., Bridger, M.: Stable Module Theory.Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence (1969). Zbl 0204.36402, MR 0269685, 10.1090/memo/0094 |
| Reference:
|
[8] Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective, and flat modules.J. Pure Appl. Algebra 210 (2007), 437-445. Zbl 1118.13014, MR 2320007, 10.1016/j.jpaa.2006.10.010 |
| Reference:
|
[9] Bennis, D., Mahdou, N.: A generalization of strongly Gorenstein projective modules.J. Algebra Appl. 8 (2009), 219-227. Zbl 1176.16008, MR 2514856, 10.1142/S021949880900328X |
| Reference:
|
[10] Christensen, L. W.: Gorenstein Dimensions.Lecture Notes in Mathematics 1747. Springer, Berlin (2000). Zbl 0965.13010, MR 1799866, 10.1007/BFb0103980 |
| Reference:
|
[11] Ding, N. Q., Chen, J. L.: The flat dimensions of injective modules.Manuscr. Math. 78 (1993), 165-177. Zbl 0804.16005, MR 1202159, 10.1007/BF02599307 |
| Reference:
|
[12] Ding, N. Q., Chen, J. L.: Coherent rings with finite self-FP-injective dimension.Commun. Algebra 24 (1996), 2963-2980. Zbl 0855.16001, MR 1396867, 10.1080/00927879608825724 |
| Reference:
|
[13] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules.Math. Z. 220 (1995), 611-633. Zbl 0845.16005, MR 1363858, 10.1007/BF02572634 |
| Reference:
|
[14] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra.de De Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2000). Zbl 0952.13001, MR 1753146, 10.1515/9783110803662 |
| Reference:
|
[15] Enochs, E. E., Jenda, O. M. G., Torrecillas, B.: Gorenstein flat modules.J. Nanjing Univ., Math. Biq. 10 (1993), 1-9. Zbl 0794.16001, MR 1248299 |
| Reference:
|
[16] Enochs, E. E., Ramos, J. A. López: Gorenstein Flat Modules.Nova Science Publishers, Huntington (2001). Zbl 1157.16300, MR 2017116 |
| Reference:
|
[17] Rozas, J. R. García, López-Ramos, J. A., Torrecillas, B.: On the existence of flat covers in $R$-$ gr$.Commun. Algebra 29 (2001), 3341-3349. Zbl 0992.16034, MR 1849490, 10.1081/AGB-100105025 |
| Reference:
|
[18] Hermann, M., Ikeda, S., Orbanz, U.: Equimultiplicity and Blowing Up. An Algebraic Study.Springer, Berlin (1988). Zbl 0649.13011, MR 0954831, 10.1007/978-3-642-61349-4 |
| Reference:
|
[19] Holm, H.: Gorenstein homological dimensions.J. Pure Appl. Algebra 189 (2004), 167-193. Zbl 1050.16003, MR 2038564, 10.1016/j.jpaa.2003.11.007 |
| Reference:
|
[20] Mao, L. X.: Strongly Gorenstein graded modules.Front. Math. China 12 (2017), 157-176. Zbl 06823674, MR 3569672, 10.1007/s11464-016-0595-y |
| Reference:
|
[21] Năstăsescu, C.: Some constructions over graded rings: Applications.J. Algebra 120 (1989), 119-138. Zbl 0678.16001, MR 0977864, 10.1016/0021-8693(89)90192-0 |
| Reference:
|
[22] Năstăsescu, C., Oystaeyen, F. Van: Graded Ring Theory.North-Holland Mathematical Library 28, North-Holland Publishing Company, Amsterdam (1982). Zbl 0494.16001, MR 0676974 |
| Reference:
|
[23] Năstăsescu, C., Oystaeyen, F. Van: Methods of Graded Rings.Lecture Notes in Mathematics 1836, Springer, Berlin (2004). Zbl 1043.16017, MR 2046303, 10.1007/b94904 |
| Reference:
|
[24] Stenström, B.: Rings of Quotients.Die Grundlehren der mathematischen Wissenschaften 217. Springer, Berlin (1975), German. Zbl 0296.16001, MR 0389953 |
| Reference:
|
[25] Yang, X., Liu, Z.: Strongly Gorenstein projective, injective and flat modules.J. Algebra 320 (2008), 2659-2674. Zbl 1173.16006, MR 2441993, 10.1016/j.jalgebra.2008.07.006 |
| Reference:
|
[26] Yang, X., Liu, Z.: FP-gr-injective modules.Math. J. Okayama Univ. 53 (2011), 83-100. Zbl 1222.16029, MR 2778885 |
| Reference:
|
[27] Zhao, G. Q., Huang, Z. Y.: $n$-strongly Gorenstein projective, injective and flat modules.Commun. Algebra 39 (2011), 3044-3062. Zbl 1247.16007, MR 2834145, 10.1080/00927872.2010.496749 |
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