THE ω-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS
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In this paper, we introduce and study the ω-weak global dimension ω-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a Pr¨ufer v-multiplication domain if and only if ω-w.gl.dim(R) ≤ 1. We also show that there is a large class of domains in which Hilbert’s syzygy Theorem for the ω-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 ω-coherent, then ω-w.gl.dim(R[x]) = ω-w.gl.dim(R).
Abstract
In this paper, we introduce and study the ω-weak global dimension ω-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a Pr¨ufer v-multiplication domain if and only if ω-w.gl.dim(R) ≤ 1. We also show that there is a large class of domains in which Hilbert’s syzygy Theorem for the ω-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 ω-coherent, then ω-w.gl.dim(R[x]) = ω-w.gl.dim(R).
참고문헌 (26)
- G. W. Chang, "Pr¨ufer ＊-multiplication domains, Nagata rings, and Kronecker function rings", J. Algebra, 319(1), pp.309~319 (2008)
- 장규환, "On characterizations of Prüfer v-multiplication domains", 한국수학논문집, 18(4), pp.335~342 (2010), 강원경기수학회
- G. W. Chang, "Integral domains in which every nonzero t-locally principal ideal is t-invertible", Comm. Algebra, 41(10), pp.3805~3819 (2013)
- S. El Baghdadi, "Ring-theoretic properties of PvMDs", Comm. Algebra, 35(5), pp.1607~1625 (2007)
- R. Gilmer, "Multiplicative Ideal Theory", Pure and Applied Mathematics, (12) (1972), Marcel Dekker, Inc
- S. Glaz, "Commutative Coherent Rings", Lecture Notes in Mathematics, 1371 (1989), Springer-Verlag
- M. Griffin, "Some results on v-multiplication rings", Canad. J. Math, 19, pp.710~722 (1967)
- E. Houston, "On t-invertibility. II", Comm. Algebra, 17(8), pp.1955~1969 (1989)
- Nv, "J. Algebra", 1989, 151~170
- B. G. Kang, "Some questions about Pr¨ufer v-multiplication domains", Comm. Algebra, 17(3), pp.553~564 (1989)
- H. Kim, "On LCM-stable modules", J. Algebra Appl, 13(4), pp.1350133~ (2014)
- T. G. Lucas, "Strong Pr¨ufer rings and the ring of finite fractions", J. Pure Appl. Algebra, 84(1), pp.59~71 (1993)
- R. Matsuda, "Notes of Pr¨ufer v-multiplication rings", Bull. Fac. Sci. Ibaraki Univ. Ser. A, (12), pp.9~15 (1980)
- J. L. Mott, "On Pr¨ufer v-multiplication domains", Manuscripta Math, 35(1-2), pp.1~26 (1981)
- Giampaolo Picozza, "A note on Prüfer semistar multiplication domains", 대한수학회지, 46(6), pp.1179~1192 (2009), 대한수학회
- P. Samuel, "Lectures on unique factorization domains, Notes by M. Pavman Murthy", Tata Institute of Fundamental Research Lectures on Mathematics, (30) (1964), Tata Institute of Fundamental Research
- F. Wang, "On w-projective modules and w-flat modules", Algebra Colloq, 4(1), pp.111~120 (1997)
- F. Wang, "w-modules over a PVMD", Proc. ISTAEM, pp.117~120 (2001)
- F. Wang, "Finitely presented type modules and w-coherent rings", J. Sichuan Normal Univ, 33, pp.1~9 (2010)
- F. Wang, "Two generalizations of projective modules and their applications", J. Pure Appl. Algebra, 219(6), pp.2099~2123 (2015)
- Fanggui Wang, "w-INJECTIVE MODULES AND w-SEMI-HEREDITARY RINGS", 대한수학회지, 51(3), pp.509~525 (2014), 대한수학회
- F. Wang, "Injective modules over w-Noetherian rings", Acta Math. Sinica (Chin. Ser.), 53(6), pp.1119~1130 (2010)
- H. Yin, "w-overrings of w-Noetherian rings", Studia Sci. Math. Hungar, 49(2), pp.200~205 (2012)
- Huayu Yin, "ω-MODULES OVER COMMUTATIVE RINGS", 대한수학회지, 48(1), pp.207~222 (2011), 대한수학회
- S. Zhao, "Flat modules over a commutative ring are w-modules", J. Sichuan Normal Univ, 35, pp.364~366 (2012)