Previous |  Up |  Next

Article

Keywords:
semidualizing module; $\mathcal {C}$-projective module; $\mathcal {C}$-(FP)-injective module; $\mathcal {C}$-flat module; noetherian ring; coherent ring
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.
References:
[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). DOI 10.1007/978-1-4684-9913-1_2 | MR 0417223 | Zbl 0301.16001
[2] Avramov, L. L., Foxby, H. B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc., III. Ser. 75 (1997), 241-270. DOI 10.1112/S0024611597000348 | MR 1455856 | Zbl 0901.13011
[3] Chase, S. U.: Direct products of modules. Trans. Am. Math. Soc. 97 (1961), 457-473. DOI 10.1090/S0002-9947-1960-0120260-3 | MR 0120260 | Zbl 0100.26602
[4] Cheatham, T. J., Stone, D. R.: Flat and projective character modules. Proc. Am. Math. Soc. 81 (1981), 175-177. DOI 10.1090/S0002-9939-1981-0593450-2 | MR 0593450 | Zbl 0458.16014
[5] Christensen, L. W.: Gorenstein Dimensions. Lecture Notes in Mathematics, vol. 1747, Springer, Berlin (2000). DOI 10.1007/BFb0103984 | MR 1799866 | Zbl 0965.13010
[6] Christensen, L. W.: Semi-dualizing complexes and their Auslander categories. Trans. Am. Math. Soc. 353 (2001), 1839-1883. DOI 10.1090/S0002-9947-01-02627-7 | MR 1813596 | Zbl 0969.13006
[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. DOI 10.1090/S0002-9947-96-01624-8 | MR 1355071 | Zbl 0862.13004
[8] Enochs, E. E., Jenda, O. M. G.: Relative homological algebra. De Gruyter Expositions in Mathematics vol. 30. Walter de Gruyter, Berlin (2000). MR 1753146 | Zbl 0952.13001
[9] Fieldhouse, D. J.: Character modules. Comment. Math. Helv. 46 (1971), 274-276. DOI 10.1007/BF02566844 | MR 0294408 | Zbl 0219.16017
[10] Foxby, H. B.: Gorenstein modules and related modules. Math. Scand. 31 (1972), 267-284. MR 0327752
[11] Glaz, S.: Commutative Coherent Rings Lecture Notes in Mathematics vol. 1371, Springer-Verlag. Berlin. (1989). MR 0999133
[12] Golod, E. S.: G-dimension and generalized perfect ideals. Proc. Steklov Inst. Math. 165 (1985), 67-71. MR 0752933 | Zbl 0589.13005
[13] Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
[14] Holm, H., Jørgensen, P.: Semi-dualizing modules and related Gorenstein homological dimensions. J. Pure Appl. Algebra. 205 (2006), 423-445. DOI 10.1016/j.jpaa.2005.07.010 | MR 2203625
[15] Holm, H., White, D.: Foxby equivalence over associative rings. J. Math. Kyoto Univ. 47 (2007), 781-808. MR 2413065 | Zbl 1154.16007
[16] Lam, T. Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics 189, Springer-Verlag, New York (1999). MR 1653294
[17] Lambek, J.: A module is flat if and only if its character module is injective. Can. Math. Bull. 7 (1964), 237-243. DOI 10.4153/CMB-1964-021-9 | MR 0163942 | Zbl 0119.27601
[18] Megibben, C.: Absolutely pure modules. Proc. Am. Math. Soc. 26 (1970), 561-566. DOI 10.1090/S0002-9939-1970-0294409-8 | MR 0294409 | Zbl 0911.16001
[19] Rotman, J. J.: An Introduction to Homological Algebra. Pure and Applied Mathematics vol. 85, Academic Press, New York (1979). MR 0538169 | Zbl 0441.18018
[20] Takahashi, R., White, D.: Homological aspects of semidualizing modules. Math. Scand. 106 (2010), 5-22. MR 2603458 | Zbl 1193.13012
[21] Vasconcelos, W. V.: Divisor Theory in Module Categories. North-Holland Mathematics Studies, II. Ser. vol. 14, Notes on Mathematica, North-Holland, Amsterdam (1974). MR 0498530 | Zbl 0296.13005
[22] White, D.: Gorenstein projective dimension with respect to a semidualizing module. J. Commutative Algebra. 2 (2010), 111-137. DOI 10.1216/JCA-2010-2-1-111 | MR 2607104
[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
[24] Zhu, X. S.: Characterize rings with character modules. Acta Math. Sinica (Chin. Ser.) 39 (1996), 743-750. MR 1443018
[25] Zhu, X. S.: Coherent rings and IF rings. Acta Math. Sin. (Chin. Ser.) 40 (1997), 845-852. MR 1612597 | Zbl 0899.16004
Partner of
EuDML logo