| Title:
|
Characterizations of some rings with $\mathcal {C}$-projective, $\mathcal {C}$-(FP)-injective and $\mathcal {C}$-flat modules (English) |
| Author:
|
Yan, Xiao Guang |
| Author:
|
Zhu, Xiao Sheng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
641-652 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative ring and $\mathcal {C}$ a semidualizing $R$-module. We investigate the relations between $\mathcal {C}$-flat modules and $\mathcal {C}$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings. (English) |
| Keyword:
|
semidualizing module |
| Keyword:
|
$\mathcal {C}$-projective module |
| Keyword:
|
$\mathcal {C}$-(FP)-injective module |
| Keyword:
|
$\mathcal {C}$-flat module |
| Keyword:
|
noetherian ring |
| Keyword:
|
coherent ring |
| MSC:
|
13C11 |
| MSC:
|
13D02 |
| MSC:
|
13D05 |
| MSC:
|
13E05 |
| MSC:
|
18G25 |
| idZBL:
|
Zbl 1249.13004 |
| idMR:
|
MR2853080 |
| DOI:
|
10.1007/s10587-011-0036-8 |
| . |
| Date available:
|
2011-09-22T14:32:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141627 |
| . |
| Reference:
|
[1] Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules.Graduate Texts in Mathematics vol. 13 New York-Heidelberg-Berlin: Springer-Verlag (1974). Zbl 0301.16001, MR 0417223, 10.1007/978-1-4684-9913-1_2 |
| Reference:
|
[2] Avramov, L. L., Foxby, H. B.: Ring homomorphisms and finite Gorenstein dimension.Proc. Lond. Math. Soc., III. Ser. 75 (1997), 241-270. Zbl 0901.13011, MR 1455856, 10.1112/S0024611597000348 |
| Reference:
|
[3] Chase, S. U.: Direct products of modules.Trans. Am. Math. Soc. 97 (1961), 457-473. Zbl 0100.26602, MR 0120260, 10.1090/S0002-9947-1960-0120260-3 |
| Reference:
|
[4] Cheatham, T. J., Stone, D. R.: Flat and projective character modules.Proc. Am. Math. Soc. 81 (1981), 175-177. Zbl 0458.16014, MR 0593450, 10.1090/S0002-9939-1981-0593450-2 |
| Reference:
|
[5] Christensen, L. W.: Gorenstein Dimensions.Lecture Notes in Mathematics, vol. 1747, Springer, Berlin (2000). Zbl 0965.13010, MR 1799866, 10.1007/BFb0103984 |
| Reference:
|
[6] Christensen, L. W.: Semi-dualizing complexes and their Auslander categories.Trans. Am. Math. Soc. 353 (2001), 1839-1883. Zbl 0969.13006, MR 1813596, 10.1090/S0002-9947-01-02627-7 |
| Reference:
|
[7] Enochs, E. E., Jenda, O. M. G., Xu, J. Z.: Foxby duality and Gorenstein injective and projective modules.Trans. Am. Math. Soc. 348 (1996), 3223-3234. Zbl 0862.13004, MR 1355071, 10.1090/S0002-9947-96-01624-8 |
| Reference:
|
[8] Enochs, E. E., Jenda, O. M. G.: Relative homological algebra.De Gruyter Expositions in Mathematics vol. 30. Walter de Gruyter, Berlin (2000). Zbl 0952.13001, MR 1753146 |
| Reference:
|
[9] Fieldhouse, D. J.: Character modules.Comment. Math. Helv. 46 (1971), 274-276. Zbl 0219.16017, MR 0294408, 10.1007/BF02566844 |
| Reference:
|
[10] Foxby, H. B.: Gorenstein modules and related modules.Math. Scand. 31 (1972), 267-284. MR 0327752, 10.7146/math.scand.a-11434 |
| Reference:
|
[11] Glaz, S.: Commutative Coherent Rings Lecture Notes in Mathematics vol. 1371, Springer-Verlag. Berlin.(1989). MR 0999133, 10.1007/BFb0084576 |
| Reference:
|
[12] Golod, E. S.: G-dimension and generalized perfect ideals.Proc. Steklov Inst. Math. 165 (1985), 67-71. Zbl 0589.13005, MR 0752933 |
| Reference:
|
[13] 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:
|
[14] Holm, H., Jørgensen, P.: Semi-dualizing modules and related Gorenstein homological dimensions.J. Pure Appl. Algebra. 205 (2006), 423-445. MR 2203625, 10.1016/j.jpaa.2005.07.010 |
| Reference:
|
[15] Holm, H., White, D.: Foxby equivalence over associative rings.J. Math. Kyoto Univ. 47 (2007), 781-808. Zbl 1154.16007, MR 2413065, 10.1215/kjm/1250692289 |
| Reference:
|
[16] Lam, T. Y.: Lectures on Modules and Rings.Graduate Texts in Mathematics 189, Springer-Verlag, New York (1999). MR 1653294 |
| Reference:
|
[17] Lambek, J.: A module is flat if and only if its character module is injective.Can. Math. Bull. 7 (1964), 237-243. Zbl 0119.27601, MR 0163942, 10.4153/CMB-1964-021-9 |
| Reference:
|
[18] Megibben, C.: Absolutely pure modules.Proc. Am. Math. Soc. 26 (1970), 561-566. Zbl 0911.16001, MR 0294409, 10.1090/S0002-9939-1970-0294409-8 |
| Reference:
|
[19] Rotman, J. J.: An Introduction to Homological Algebra.Pure and Applied Mathematics vol. 85, Academic Press, New York (1979). Zbl 0441.18018, MR 0538169 |
| Reference:
|
[20] Takahashi, R., White, D.: Homological aspects of semidualizing modules.Math. Scand. 106 (2010), 5-22. Zbl 1193.13012, MR 2603458, 10.7146/math.scand.a-15121 |
| Reference:
|
[21] Vasconcelos, W. V.: Divisor Theory in Module Categories.North-Holland Mathematics Studies, II. Ser. vol. 14, Notes on Mathematica, North-Holland, Amsterdam (1974). Zbl 0296.13005, MR 0498530 |
| Reference:
|
[22] White, D.: Gorenstein projective dimension with respect to a semidualizing module.J. Commutative Algebra. 2 (2010), 111-137. MR 2607104, 10.1216/JCA-2010-2-1-111 |
| Reference:
|
[23] Sather-Wagstaff, S., Sharif, T., White, D.: AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules.(to appear) in Algebr. Represent. Theor. MR 2785915 |
| Reference:
|
[24] Zhu, X. S.: Characterize rings with character modules.Acta Math. Sinica (Chin. Ser.) 39 (1996), 743-750. MR 1443018 |
| Reference:
|
[25] Zhu, X. S.: Coherent rings and IF rings.Acta Math. Sin. (Chin. Ser.) 40 (1997), 845-852. Zbl 0899.16004, MR 1612597 |
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