MAX-INJECTIVE, MAX-FLAT MODULES AND MAX-COHERENT RINGS
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초록
A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. MI (resp. MF) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an MF-preenvelope, and every left R-module has an MI-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic MI-cover if and only if any right R-module has a monic MF-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, MI-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.
Abstract
A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. MI (resp. MF) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an MF-preenvelope, and every left R-module has an MI-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic MI-cover if and only if any right R-module has a monic MF-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, MI-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.
참고문헌 (15)
- F. W. Anderson, "Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13" (1974), Springer-Verlag
- L. Bican, "All modules have flat covers", Bull. London Math. Soc., 33(4), pp.385~390 (2001)
- N. Q. Ding, "On envelopes with the unique mapping property", Comm. Algebra, 24(4), pp.1459~1470 (1996)
- E. E. Enochs, "Injective and flat covers, envelopes and resolvents", Israel J. Math., 39(3), pp.189~209 (1981)
- E. E. Enochs, "Relative Homological Algebra" (2000), Walter de Gruyter & Co.
- C. Faith, "Algebra II, Grundlehren der Mathematischen Wissenschaften, No. 191" (1976), Springer-Verlag
- D. J. Fieldhouse, "Pure theories", Math. Ann., 184, pp.1~18 (1969)
- L. Fuchs, "Modules over Non-Noetherian Domains" (2001), American Mathematical Society
- L. X. Mao, "Min-flat modules and min-coherent rings", Comm. Algebra, 35(2), pp.635~650 (2007)
- W. K. Nicholson, "Quasi-Frobenius Rings" (2003), Cambridge University Press
- J. J. Rotman, "An Introduction to Homological Algebra" (1979), Academic Press, Inc.
- J. Rada, "Rings characterized by (pre)envelopes and (pre)covers of their modules", Comm. Algebra, 26(3), pp.899~912 (1998)
- M. Y. Wang, "Frobenius Structure in Algebra (Chinese)" (2005), Science Press
- M. Y. Wang, "On maximal injectivity", Acta Math. Sin. (Engl. Ser.), 21(6), pp.1451~1458 (2005)
- J. Z. Xu, "Flat Covers of Modules, Lecture Notes in Mathematics, 1634", Springer-Verlag