MAX-INJECTIVE, MAX-FLAT MODULES AND MAX-COHERENT RINGS

Title & Authors
MAX-INJECTIVE, MAX-FLAT MODULES AND MAX-COHERENT RINGS
Xiang, Yueming;

Abstract
A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. $\small{\mathfrak{M}\mathfrak{I}}$ (resp. $\small{\mathfrak{M}\mathfrak{F}}$) 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 $\small{\mathfrak{M}\mathfrak{F}}$-preenvelope, and every left R-module has an $\small{\mathfrak{M}\mathfrak{I}}$-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic $\small{\mathfrak{M}\mathfrak{I}}$-cover if and only if any right R-module has a monic $\small{\mathfrak{M}\mathfrak{F}}$-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, $\small{\mathfrak{M}\mathfrak{I}}$-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.
Keywords
max-injective (pre)cover;max-flat preenvelope;max-coherent ring;MI-injective module;MI-flat module;$\small{\mathfrak{M}\mathfrak{I}}$-dimension;
Language
English
Cited by
1.
Neat-flat Modules, Communications in Algebra, 2016, 44, 1, 416
2.
Absolutelys-Pure Modules and Neat-Flat Modules, Communications in Algebra, 2015, 43, 2, 384
References
1.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13. Springer-Verlag, New York-Heidelberg, 1974.

2.
L. Bican, R. El Bashir, and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385-390.

3.
N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470.

4.
E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209.

5.
E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter & Co., Berlin, 2000.

6.
C. Faith, Algebra II, Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag, Berlin-New York, 1976.

7.
D. J. Fieldhouse, Pure theories, Math. Ann. 184 (1969), 1-18.

8.
L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84. American Mathematical Society, Providence, RI, 2001.

9.
L. X. Mao, Min-flat modules and min-coherent rings, Comm. Algebra 35 (2007), no. 2, 635-650.

10.
W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics, 158. Cambridge University Press, Cambridge, 2003.

11.
J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

12.
J. Rada and M. Saorin, Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26 (1998), no. 3, 899-912.

13.
M. Y. Wang, Frobenius Structure in Algebra (Chinese), Science Press, Beijing, 2005.

14.
M. Y. Wang and G. Zhao, On maximal injectivity, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1451-1458.

15.
J. Z. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, 1634. Springer-Verlag, Berlin, 1996.