乘法封闭集的伪局部化与PVMR的刻画
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摘要
近年来关于PVMR的研究见诸于不少文献,一直受到人们的关注.本文运用伪局部化方法与一般交换环上w-算子刻画PVMR.第一章首先引入伪局部化R[S]与S-可约,并探讨了R[S]的一些基本性质.然后,将一般交换环上的w-算子与伪局部化R[S]相结合,得到模的伪局部整体原理.第二章首先介绍了Manis赋值理论,定义了伪赋值环,即P是R的唯一正则极大理想且(R,P)是赋值对.并结合Manis赋值理论与伪局部整体原理,得到PVMR的等价刻画,即R是PVMR当且仅当对R的任何的极大w-理想Q, (R[Q],[Q]R[Q])是Manis赋值环;这也等价于对于R的任何的正则极大w-理想P, (R[P],[P]R[P])是伪赋值环.其次,采用传统的理想理论的研究方法,结合w-算子得到PVMR的另一等价刻画,即R是PVMR当且仅当R的每个有限型的正则理想是w-平坦理想(w-投射理想);若R是弱DW环,则R是PVMR当且仅当R是Pru|¨fer环.
PVMRs have received a good deal of attention continuously in a number ofliterature in recent decade. In this thesis, PVMRs are characterized by utiliz-ing pseudo-localization principle and w-operations. In chapter 1, the concept ofpseudo-localization R[S] and S-cancellation are introduced. Then, we obtain someproperties of pseudo-localization R[S]. Moreover, the pseudo-local to global prin-ciple is received by using pseudo-localization R[S] and w-operations. In chapter2, Manis valuation theory is presented. Pseudo-valuation ring R is defined, whichhave a unique regular maximal ideal P and (R,P) is a valuation pair. On theone hand, by using pseudo-localization principle and w-operations, it is shownthat R is a PVMR if and only if (R[Q],[Q]R[Q]) is a Manis valuation ring for anymaximal w-ideal Q of R; which is equivalent to that (R[P],[P]R[P]) is a pseudo-valuation ring for any regular maximal w-ideal P of R. On the other hand, byusing traditional ideal theory methods, we proved that R is a PVMR if and onlyif every finite type ideal of R is w-projective(w-flat). If R is a weak DW ring,then R is a PVMR if and only if R is a Pru|¨fer ring.
PVMRs have received a good deal of attention continuously in a number ofliterature in recent decade. In this thesis, PVMRs are characterized by utiliz-ing pseudo-localization principle and w-operations. In chapter 1, the concept ofpseudo-localization R[S] and S-cancellation are introduced. Then, we obtain someproperties of pseudo-localization R[S]. Moreover, the pseudo-local to global prin-ciple is received by using pseudo-localization R[S] and w-operations. In chapter2, Manis valuation theory is presented. Pseudo-valuation ring R is defined, whichhave a unique regular maximal ideal P and (R,P) is a valuation pair. On theone hand, by using pseudo-localization principle and w-operations, it is shownthat R is a PVMR if and only if (R[Q],[Q]R[Q]) is a Manis valuation ring for anymaximal w-ideal Q of R; which is equivalent to that (R[P],[P]R[P]) is a pseudo-valuation ring for any regular maximal w-ideal P of R. On the other hand, byusing traditional ideal theory methods, we proved that R is a PVMR if and onlyif every finite type ideal of R is w-projective(w-flat). If R is a weak DW ring,then R is a PVMR if and only if R is a Pru|¨fer ring.
引文
[1] YIN Hua-yu, WANG Fang-gui, ZHU Xiao-shen, CHEN You-hua. w-Modulesover Commutative rings[J]. to appear.
[2] R. Gilmer. Multiplicative Ideal Theory. New York: Marcel Dekker INC, 1972.
[3] L. Fuchs, L. Salce. Modules over Valuation Domains. New York: MarcelDekker INC, 1985.
[4] J.L. Mott, M. Zafrullah. On Pru¨fer v-multiplication domains. ManuscriptaMath., 1981, 35: 1–26.
[5] WANG Fanggui, R.L. McCasland. On w-modules over Strong Mori domains.Comm. Algebra, 1997, 25(4): 1285–1306.
[6] J.R. Hedstrom, E.G. Houston. Some remarks on star-operations. J. Pure Appl.Algebra, 1980, 18: 37–44.
[7] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge UniversityPress, 1980.
[8] WANG Fanggui. w-?at Modules over Commutative Rings. to appear.
[9] M. Zafrullah. Flatness and invertibility of an ideal. Comm. Algebra, 1990,18(7): 2151–2158.
[10] WANG Fanggui. w-Dimension of domains. Comm. Algebra, 1999, 27(5):2267–2276.
[11] WANG Fanggui. w-Dimension of domains II. Comm. Algebra, 2001, 29(6):2419–2428.
[12] WANG Fanggui. On w-projective modules and w-?at modules. Algebra Col-loq., 1997, 4: 111–120.
[13] WANG Fanggui. On w-coherent Domains. to appear.
[14] Malcolm Gri?n. On Pru¨fer rings with zero-divisors J.Reine AngewMath.239-240(1970),55-67.
[15] D.E. Dobbs, E.G. Houston, T.G. Lucas and M. Zafrullah. t-linked overringsand Pru¨fer v-multiplication domains. Comm. Algebra, 1989, 17(11): 2835–2852.
[16] D.E. Dobbs, E.G. Houston, T.G. Lucas, M. Roitman and M. Zafrullah. Ont-linked overrings. Comm. Algebra, 1992, 20(5): 1463–1488.
[17] WANG Fanggui. w-projective Modules over Commutative Rings. to appear.
[18] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics, no.1371. Berlin and New York, Springer-Verlag, 1989.
[19] WANG Fanggui, R.L. McCasland. On strong Mori domains. J. Pure Appl.Algebra, 1999, 135: 155–165.
[20] J.D. Sally, W.V. Vasconcelos. Flat ideals I. Comm. Algebra, 1975, 3(6): 531–543.
[21] WANG Fanggui. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117–120.
[22] J.J. Rotman. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[23] WANG Fanggui. On induced operations and UMT-domains. J. Sichuan Nor-mal Univ.(Natural Science), 2004, 27(1): 1–9.
[24] V. Barucci, S. Gabelli. On semi-Krull domains. J. Algebra, 1992, 145: 306–328.
[25] B.G. Kang. On the converse of a well-known fact about Krull domains. J.Algebra, 1989, 124: 284–299.
[26] M.H. Park. Group rings and semigroup rings over strong Mori domains. J.Pure Appl. Algebra, 2001, 163: 301–318.
[27] E. Houston, M. Zafrullah. Integral domains in which each t-ideal is divisorial.Michigan Math. J., 1988, 35: 291–300.
[28] J.L. Mott, M. Zafrullah. On Krull domains. Arch. Math., 1991, 56: 559–568.
[29] E. Kunz. Introduction to Commutative Algebra and Algebraic Geometry.Boston: Birkh¨auser, 1985.
[30] J.W. Brewer, W.J. Heinzer. Associated primes of principal ideals. DukeMath. J., 1974, 41: 1–7.
[31] M. Zafrullah. On generalized Dedekind domains. Mathematika, 1986, 33:285–295.
[32] S. Malik, J.L. Mott, M. Zafrullah. On t-invertibility. Comm. Algebra, 1988,16(1): 149–170.
[33] B.G. Kang. Pru¨fer v-multiplication domains and the ring R[X]Nv. J. Algebra,1989, 123: 151–170.
[34] WANG Fanggui. On t-dimension over strong Mori domains. Acta Mathe-matica Sinica, English Series, 2006, 22(1): 131–138.
[35] Monte B. Boisen, JR. And Max D.Larsen.Pru¨fer and Valuation Rings withzero divisors.Pacific Journal of Mathmatics, Vol.40.No.1.1972
[36] D.D. Anderson, B.G. Kang. Pseudo-Dedekind domains and divisorial idealsin R[X]T. J. Algebra, 1989, 122: 323–336.
[37] E. Houston, M. Zafrullah. On t-invertibility II. Comm. Algebra, 1989, 17(8):1955–1969.
[38] Patrick H. Kelly,and Max D. larsen. Maximal partial homomorphisms andvaluation pairs.Proc. Amer.Math.Soc.30(1971),426-430.
[39] M.E.Mains Extension of Valuation Theory.Bull.Am.Math. Soc.78(1967),735-736.
[40] W.Krull. Beitr¨age zur Arithmetik Commutative Integrit¨atsbereiche. Math.zeit. 41(1936)545-577.
[41] Silvana Bazzoni, Sarah Glaz. Pru¨fer rings. to apprar.
[42]王芳贵.交换环与星型算子理论,科学出版社, 2006年.
[43]王芳贵.交换环上的w-模理论.讲义.
[44]王芳贵. GCD整环与UFD分式理想的刻划.科学通报, 1997, 42(5): 556.
[45]陈幼华,王芳贵,尹华玉.关于PVMD的一些刻画.四川师范大学学报(自然科学版). 2008, 31(1)18-21.
[2] R. Gilmer. Multiplicative Ideal Theory. New York: Marcel Dekker INC, 1972.
[3] L. Fuchs, L. Salce. Modules over Valuation Domains. New York: MarcelDekker INC, 1985.
[4] J.L. Mott, M. Zafrullah. On Pru¨fer v-multiplication domains. ManuscriptaMath., 1981, 35: 1–26.
[5] WANG Fanggui, R.L. McCasland. On w-modules over Strong Mori domains.Comm. Algebra, 1997, 25(4): 1285–1306.
[6] J.R. Hedstrom, E.G. Houston. Some remarks on star-operations. J. Pure Appl.Algebra, 1980, 18: 37–44.
[7] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge UniversityPress, 1980.
[8] WANG Fanggui. w-?at Modules over Commutative Rings. to appear.
[9] M. Zafrullah. Flatness and invertibility of an ideal. Comm. Algebra, 1990,18(7): 2151–2158.
[10] WANG Fanggui. w-Dimension of domains. Comm. Algebra, 1999, 27(5):2267–2276.
[11] WANG Fanggui. w-Dimension of domains II. Comm. Algebra, 2001, 29(6):2419–2428.
[12] WANG Fanggui. On w-projective modules and w-?at modules. Algebra Col-loq., 1997, 4: 111–120.
[13] WANG Fanggui. On w-coherent Domains. to appear.
[14] Malcolm Gri?n. On Pru¨fer rings with zero-divisors J.Reine AngewMath.239-240(1970),55-67.
[15] D.E. Dobbs, E.G. Houston, T.G. Lucas and M. Zafrullah. t-linked overringsand Pru¨fer v-multiplication domains. Comm. Algebra, 1989, 17(11): 2835–2852.
[16] D.E. Dobbs, E.G. Houston, T.G. Lucas, M. Roitman and M. Zafrullah. Ont-linked overrings. Comm. Algebra, 1992, 20(5): 1463–1488.
[17] WANG Fanggui. w-projective Modules over Commutative Rings. to appear.
[18] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics, no.1371. Berlin and New York, Springer-Verlag, 1989.
[19] WANG Fanggui, R.L. McCasland. On strong Mori domains. J. Pure Appl.Algebra, 1999, 135: 155–165.
[20] J.D. Sally, W.V. Vasconcelos. Flat ideals I. Comm. Algebra, 1975, 3(6): 531–543.
[21] WANG Fanggui. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117–120.
[22] J.J. Rotman. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[23] WANG Fanggui. On induced operations and UMT-domains. J. Sichuan Nor-mal Univ.(Natural Science), 2004, 27(1): 1–9.
[24] V. Barucci, S. Gabelli. On semi-Krull domains. J. Algebra, 1992, 145: 306–328.
[25] B.G. Kang. On the converse of a well-known fact about Krull domains. J.Algebra, 1989, 124: 284–299.
[26] M.H. Park. Group rings and semigroup rings over strong Mori domains. J.Pure Appl. Algebra, 2001, 163: 301–318.
[27] E. Houston, M. Zafrullah. Integral domains in which each t-ideal is divisorial.Michigan Math. J., 1988, 35: 291–300.
[28] J.L. Mott, M. Zafrullah. On Krull domains. Arch. Math., 1991, 56: 559–568.
[29] E. Kunz. Introduction to Commutative Algebra and Algebraic Geometry.Boston: Birkh¨auser, 1985.
[30] J.W. Brewer, W.J. Heinzer. Associated primes of principal ideals. DukeMath. J., 1974, 41: 1–7.
[31] M. Zafrullah. On generalized Dedekind domains. Mathematika, 1986, 33:285–295.
[32] S. Malik, J.L. Mott, M. Zafrullah. On t-invertibility. Comm. Algebra, 1988,16(1): 149–170.
[33] B.G. Kang. Pru¨fer v-multiplication domains and the ring R[X]Nv. J. Algebra,1989, 123: 151–170.
[34] WANG Fanggui. On t-dimension over strong Mori domains. Acta Mathe-matica Sinica, English Series, 2006, 22(1): 131–138.
[35] Monte B. Boisen, JR. And Max D.Larsen.Pru¨fer and Valuation Rings withzero divisors.Pacific Journal of Mathmatics, Vol.40.No.1.1972
[36] D.D. Anderson, B.G. Kang. Pseudo-Dedekind domains and divisorial idealsin R[X]T. J. Algebra, 1989, 122: 323–336.
[37] E. Houston, M. Zafrullah. On t-invertibility II. Comm. Algebra, 1989, 17(8):1955–1969.
[38] Patrick H. Kelly,and Max D. larsen. Maximal partial homomorphisms andvaluation pairs.Proc. Amer.Math.Soc.30(1971),426-430.
[39] M.E.Mains Extension of Valuation Theory.Bull.Am.Math. Soc.78(1967),735-736.
[40] W.Krull. Beitr¨age zur Arithmetik Commutative Integrit¨atsbereiche. Math.zeit. 41(1936)545-577.
[41] Silvana Bazzoni, Sarah Glaz. Pru¨fer rings. to apprar.
[42]王芳贵.交换环与星型算子理论,科学出版社, 2006年.
[43]王芳贵.交换环上的w-模理论.讲义.
[44]王芳贵. GCD整环与UFD分式理想的刻划.科学通报, 1997, 42(5): 556.
[45]陈幼华,王芳贵,尹华玉.关于PVMD的一些刻画.四川师范大学学报(自然科学版). 2008, 31(1)18-21.