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http://dx.doi.org/10.4134/BKMS.2015.52.4.1327

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS  

WANG, FANGGUI (COLLEGE OF MATHEMATICS AND SOFTWARE SCIENCE SICHUAN NORMAL UNIVERSITY)
QIAO, LEI (COLLEGE OF MATHEMATICS AND SOFTWARE SCIENCE SICHUAN NORMAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.4, 2015 , pp. 1327-1338 More about this Journal
Abstract
In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).
Keywords
GV-torsionfree module; w-module; w-flat module; w-flat dimension; w-weak global dimension;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 F. Wang, w-modules over a PVMD, Proc. ISTAEM, Hong Kong, 117-120, 2001.
2 F. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ. 33 (2010), 1-9.
3 F. Wang and H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra 219 (2015), no. 6, 2099-2123.   DOI   ScienceOn
4 F. Wang and H. Kim, w-injective modules and w-semi-hereditary rings, J. Korean Math. Soc. 51 (2014), no. 3, 509-525.   DOI   ScienceOn
5 F. Wang and J. Zhang, Injective modules over w-Noetherian rings, Acta Math. Sinica (Chin. Ser.) 53 (2010), no. 6, 1119-1130.
6 H. Yin and Y. Chen, w-overrings of w-Noetherian rings, Studia Sci. Math. Hungar. 49 (2012), no. 2, 200-205.   DOI
7 H. Yin, F. Wang, X. Zhu, and Y. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 207-222.   DOI   ScienceOn
8 S. Zhao, F. Wang, and H. Chen, Flat modules over a commutative ring are w-modules, J. Sichuan Normal Univ. 35 (2012), 364-366.
9 G. W. Chang, Prufer *-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra 319 (2008), no. 1, 309-319.   DOI   ScienceOn
10 G. W. Chang, On characterizations of Prufer v-multiplication domains, Korean J. Math. 18 (2010), no. 4, 335-342.
11 G. W. Chang, H. Kim, and J. W. Lim, Integral domains in which every nonzero t-locally principal ideal is t-invertible, Comm. Algebra 41 (2013), no. 10, 3805-3819.   DOI
12 S. El Baghdadi and S. Gabelli, Ring-theoretic properties of PvMDs, Comm. Algebra 35 (2007), no. 5, 1607-1625.   DOI   ScienceOn
13 R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, No. 12. Marcel Dekker, Inc., New York, 1972.
14 S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371. Springer-Verlag, Berlin, 1989.
15 B. G. Kang, Some questions about Prufer v-multiplication domains, Comm. Algebra 17 (1989), no. 3, 553-564.   DOI   ScienceOn
16 M. Griffin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722.   DOI
17 E. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969.   DOI   ScienceOn
18 B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), no. 1, 151-170.   DOI
19 H. Kim and F. Wang, On LCM-stable modules, J. Algebra Appl. 13 (2014), no. 4, 1350133, 18 pp.
20 T. G. Lucas, Strong Prufer rings and the ring of finite fractions, J. Pure Appl. Algebra 84 (1993), no. 1, 59-71.   DOI   ScienceOn
21 R. Matsuda, Notes of Prufer v-multiplication rings, Bull. Fac. Sci. Ibaraki Univ. Ser. A No. 12 (1980), 9-15.
22 J. L. Mott and M. Zafrullah, On Prufer v-multiplication domains, Manuscripta Math. 35 (1981), no. 1-2, 1-26.   DOI
23 G. Picozza, A note on Prufer semistar multiplication domains, J. Korean Math. Soc. 46 (2009), no. 6, 1179-1192.   DOI   ScienceOn
24 J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
25 P. Samuel, Lectures on unique factorization domains, Notes by M. Pavman Murthy, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30. Tata Institute of Fundamental Research, Bombay, 1964.
26 F. Wang, On w-projective modules and w-flat modules, Algebra Colloq. 4 (1997), no. 1, 111-120.