ω~T-Noether环与μ-Noether环
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摘要
本文首先在交换环中引入gυ-理想、gυ-无挠模、ω~T-模,模的ω~T-包络;给出了ω~T-Noether环的定义.我们讨论了ω~T-Noether环的相伴素理想和准素分解.通过引入ω~T-不可约理想,证明了ω~T-Noether环上的任何真ω~T-理想都有准素ω~T-理想的准素分解.通过研究ω~T-Noether环上内射模的结构和性质,将Cartan-Eilenberg-Bass定理推广到了ω~T-Noether环上: R是ω~T-Noether环当且仅当任意多个gυ-无挠内射模的直和是内射模,当且仅当每个gυ-无挠内射模是Σ-内射模.我们还证明了若R是ω~T-Noether环,I是真ω~T-理想,则E(R/I)可以写成有限个不可分解内射模的直和.最后,我们引入了υ-Noether环的定义(具有υ-理想的升链条件的交换环).证明了每个ω~T-Noether环都是υ-Noether环;证明了若P为R中的素理想, R是υ-Noether环,则R[P]也是υ-Noether环;还证明了R中每个非零理想都被包含在至多有限个极大t-理想中,如果对于每个极大t-理想M而言, R[M]是υ-Noether环,则R是υ-Noether环.
In this paper, we introduce gυ-ideal, gυ-torsion-free module,ω~T-module,ω~T-envelope in Commutative rings and defineω~T-Noetherian ring. We discuss as-sociate prime ideals and primary decomposition ofω~T-Noetherian rings. Byintroducingω~T-irreducible ideal, we prove that any properω~T-ideal has primarydecomposition of primaryω~T-ideal. By studying the structure and propertiesof injective modules ofω~T-Noetherian rings, we extend Cartan-Eilenberg-BassTheorem toω~T-Noetherian rings: R is aω~T-Noetherian ring if and only if everydirect sum of gv -torsion-free injective modules is injective; if and only if everygv -torsion- free injective module isΣ-injective. Let R be aω~T-Noetherian ring,we also prove that E(R/I) can be presented finite direct sum of indecomposableinjective modules if I is a properω~T-ideal. Finally, we defineυ-Noetherian ring(ie. having the condition of ascending chain onυ-ideals). We prove that everyω~T-Noetherian ring is aυ-Noetherian ring. Let R be aυ-Noetherian ring, weprove that R[P] is aυ-Noetherian ring if P is a prime ideal. We also prove that ifeach nonzero ideal is contained in at most finitely many maximal t-ideals and foreach maximal t-ideal M, R[M] is aυ-Noetherian ring, then R is aυ-Noetherianring.
In this paper, we introduce gυ-ideal, gυ-torsion-free module,ω~T-module,ω~T-envelope in Commutative rings and defineω~T-Noetherian ring. We discuss as-sociate prime ideals and primary decomposition ofω~T-Noetherian rings. Byintroducingω~T-irreducible ideal, we prove that any properω~T-ideal has primarydecomposition of primaryω~T-ideal. By studying the structure and propertiesof injective modules ofω~T-Noetherian rings, we extend Cartan-Eilenberg-BassTheorem toω~T-Noetherian rings: R is aω~T-Noetherian ring if and only if everydirect sum of gv -torsion-free injective modules is injective; if and only if everygv -torsion- free injective module isΣ-injective. Let R be aω~T-Noetherian ring,we also prove that E(R/I) can be presented finite direct sum of indecomposableinjective modules if I is a properω~T-ideal. Finally, we defineυ-Noetherian ring(ie. having the condition of ascending chain onυ-ideals). We prove that everyω~T-Noetherian ring is aυ-Noetherian ring. Let R be aυ-Noetherian ring, weprove that R[P] is aυ-Noetherian ring if P is a prime ideal. We also prove that ifeach nonzero ideal is contained in at most finitely many maximal t-ideals and foreach maximal t-ideal M, R[M] is aυ-Noetherian ring, then R is aυ-Noetherianring.
引文
[1] WANG Fanggui, R.L. McCasland. On w-modules over Strong Mori domains.Comm. Algebra, 1997, 25(4): 1285–1306.
[2] WANG Fanggui, R.L. McCasland. On strong Mori domains. J. Pure Appl.Algebra, 1999, 135: 155–165.
[3] A.Mimouni. TW-domains and Strong Mori domains.J.Pure Appl. Algebra,2003, 177:79-93.
[4] E.Matlis. Injective modules over Noetherian rings. Pac.J.Math, 1958, 6:512-528.
[5] T.Lucas. The mori property in rings with zero divisors. Lecture Notes in PureAppl.Math, 2004, 236:379-400.
[6] T.Lucas. Krull rings,Pru¨fer v-mutiplication rings and The ring of finite frac-tions. Rocky Mountain Journal of Mathmatics , 2005, 35(4):1251-1325.
[7] I.Beck. Injective modules over a Krull domain . J. Algebra, 1971, 17:116-131.
[8] I.Beck. ?Injective modules. J. Algebra, 1972, 21:232-249.
[9] Said El Baghdadi, Stefania Gabelli. w-Divisorial domains. J. Algebra,2005,285:335-355.
[10] L.Fuchs. Modules over Mori domains. Studia Scientiarum MathematicarumHungarica, 2005, 40:33-40.
[11] J.Huckaba. Commutative Rings with zero divisors. Dekker,New York, 1988.
[12] M.Zafrullah. Ascending chain conditions and Star operation . Comm Alge-bra, 1989, 17(6):1523-1533.
[13] B.G.Kang. Characterizations of Krull rings with zero divisors. J.PureAppl.Algebra, 2000, 146:283-290.
[14]王芳贵.交换环与星型算子理论.科学出版社, 2006.
[15] WANG Fanggui,ZHANG Jun. Injective modules over w-Noetherian rings. toappear.
[16] R. Gilmer. Multiplicative Ideal Theory. New York: Marcel Dekker INC, 1972.
[17] L. Fuchs, L. Salce. Modules over Valuation Domains. New York: MarcelDekker INC, 1985.
[18] J.L. Mott, M. Zafrullah. On Pru¨fer v-multiplication domains. ManuscriptaMath., 1981, 35: 1–26.
[19] J.R. Hedstrom, E.G. Houston. Some remarks on star-operations. J. PureAppl. Algebra, 1980, 18: 37–44.
[20] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge Univer-sity Press, 1980.
[21] D.D. Anderson, M. Zafrullah. On t-invertibility III. Comm. Algebra, 1993,21(4): 1189–1201.
[22] M. Zafrullah. Flatness and invertibility of an ideal. Comm. Algebra, 1990,18(7): 2151–2158.
[23] WANG Fanggui. w-Dimension of domains. Comm. Algebra, 1999, 27(5):2267–2276.
[24] WANG Fanggui. w-Dimension of domains II. Comm. Algebra, 2001, 29(6):2419–2428.
[25] WANG Fanggui. On w-projective modules and w-?at modules. Algebra Col-loq., 1997, 4: 111–120.
[26] 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.
[27] D.E. Dobbs, E.G. Houston, T.G. Lucas, M. Roitman and M. Zafrullah. Ont-linked overrings. Comm. Algebra, 1992, 20(5): 1463–1488.
[28] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics, no.1371. Berlin and New York, Springer-Verlag, 1989.
[29] Gyu WhanChang .Strong Mori domains and the ring D[X]Nv. J.PureAppl.Algebra,2005,197:293-304.
[30] J.D. Sally, W.V. Vasconcelos. Flat ideals I. Comm. Algebra, 1975, 3(6): 531–543.
[31] WANG Fanggui. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117–120.
[32] J.J. Rotman. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[33] WANG Fanggui. On induced operations and UMT-domains. J. Sichuan Nor-mal Univ.(Natural Science), 2004, 27(1): 1–9.
[34] M.H. Park. Group rings and semigroup rings over strong Mori domains. J.Pure Appl. Algebra, 2001, 163: 301–318.
[35] E. Houston, M. Zafrullah. Integral domains in which each t-ideal is divisorial.Michigan Math. J., 1988, 35: 291–300.
[36] J.L. Mott, M. Zafrullah. On Krull domains. Arch. Math., 1991, 56: 559–568.
[37] E. Kunz. Introduction to Commutative Algebra and Algebraic Geometry.Boston: Birkh¨auser, 1985.
[38] J.W. Brewer, W.J. Heinzer. Associated primes of principal ideals. DukeMath. J., 1974, 41: 1–7.
[39] S. Malik, J.L. Mott, M. Zafrullah. On t-invertibility. Comm. Algebra, 1988,16(1): 149–170.
[40] B.G. Kang. Pru¨fer v-multiplication domains and the ring R[X]Nv. J. Algebra,1989, 123: 151–170.
[41]王芳贵. GCD整环与UFD分式理想的刻划.科学通报, 1997, 42(5): 556.
[42] WANG Fanggui. On t-dimension over strong Mori domains. Acta Mathe-matica Sinica, English Series, 2006, 22(1): 131–138.
[43] N. Jacobson. Basic Algebra II. San Francisco: W.H. Freeman, 1980.
[44] E.G. Houston, T.G. Lucas, T.M. Viswanathan. Primary decomposition ofdivisorial ideals in Mori domains. J. Algebra, 1988, 117: 327–342.
[45] D.D. Anderson, B.G. Kang. Pseudo-Dedekind domains and divisorial idealsin R[X]T. J. Algebra, 1989, 122: 323–336.
[46] E. Houston, M. Zafrullah. On t-invertibility II. Comm. Algebra, 1989, 17(8):1955–1969.
[47] V. Barucci, D.F. Anderson, D.E. Dobbs. Coherent Mori domains and theprincipal ideal theorem. Comm. Algebra, 1987, 15(6): 1119–1156.
[48] M. Auslander, D.A. Buchsbaum. Homological dimension in Noetherian ringsII. Trans. Amer. Math. Soc., 1958, 88: 194–206.
[2] WANG Fanggui, R.L. McCasland. On strong Mori domains. J. Pure Appl.Algebra, 1999, 135: 155–165.
[3] A.Mimouni. TW-domains and Strong Mori domains.J.Pure Appl. Algebra,2003, 177:79-93.
[4] E.Matlis. Injective modules over Noetherian rings. Pac.J.Math, 1958, 6:512-528.
[5] T.Lucas. The mori property in rings with zero divisors. Lecture Notes in PureAppl.Math, 2004, 236:379-400.
[6] T.Lucas. Krull rings,Pru¨fer v-mutiplication rings and The ring of finite frac-tions. Rocky Mountain Journal of Mathmatics , 2005, 35(4):1251-1325.
[7] I.Beck. Injective modules over a Krull domain . J. Algebra, 1971, 17:116-131.
[8] I.Beck. ?Injective modules. J. Algebra, 1972, 21:232-249.
[9] Said El Baghdadi, Stefania Gabelli. w-Divisorial domains. J. Algebra,2005,285:335-355.
[10] L.Fuchs. Modules over Mori domains. Studia Scientiarum MathematicarumHungarica, 2005, 40:33-40.
[11] J.Huckaba. Commutative Rings with zero divisors. Dekker,New York, 1988.
[12] M.Zafrullah. Ascending chain conditions and Star operation . Comm Alge-bra, 1989, 17(6):1523-1533.
[13] B.G.Kang. Characterizations of Krull rings with zero divisors. J.PureAppl.Algebra, 2000, 146:283-290.
[14]王芳贵.交换环与星型算子理论.科学出版社, 2006.
[15] WANG Fanggui,ZHANG Jun. Injective modules over w-Noetherian rings. toappear.
[16] R. Gilmer. Multiplicative Ideal Theory. New York: Marcel Dekker INC, 1972.
[17] L. Fuchs, L. Salce. Modules over Valuation Domains. New York: MarcelDekker INC, 1985.
[18] J.L. Mott, M. Zafrullah. On Pru¨fer v-multiplication domains. ManuscriptaMath., 1981, 35: 1–26.
[19] J.R. Hedstrom, E.G. Houston. Some remarks on star-operations. J. PureAppl. Algebra, 1980, 18: 37–44.
[20] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge Univer-sity Press, 1980.
[21] D.D. Anderson, M. Zafrullah. On t-invertibility III. Comm. Algebra, 1993,21(4): 1189–1201.
[22] M. Zafrullah. Flatness and invertibility of an ideal. Comm. Algebra, 1990,18(7): 2151–2158.
[23] WANG Fanggui. w-Dimension of domains. Comm. Algebra, 1999, 27(5):2267–2276.
[24] WANG Fanggui. w-Dimension of domains II. Comm. Algebra, 2001, 29(6):2419–2428.
[25] WANG Fanggui. On w-projective modules and w-?at modules. Algebra Col-loq., 1997, 4: 111–120.
[26] 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.
[27] D.E. Dobbs, E.G. Houston, T.G. Lucas, M. Roitman and M. Zafrullah. Ont-linked overrings. Comm. Algebra, 1992, 20(5): 1463–1488.
[28] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics, no.1371. Berlin and New York, Springer-Verlag, 1989.
[29] Gyu WhanChang .Strong Mori domains and the ring D[X]Nv. J.PureAppl.Algebra,2005,197:293-304.
[30] J.D. Sally, W.V. Vasconcelos. Flat ideals I. Comm. Algebra, 1975, 3(6): 531–543.
[31] WANG Fanggui. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117–120.
[32] J.J. Rotman. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[33] WANG Fanggui. On induced operations and UMT-domains. J. Sichuan Nor-mal Univ.(Natural Science), 2004, 27(1): 1–9.
[34] M.H. Park. Group rings and semigroup rings over strong Mori domains. J.Pure Appl. Algebra, 2001, 163: 301–318.
[35] E. Houston, M. Zafrullah. Integral domains in which each t-ideal is divisorial.Michigan Math. J., 1988, 35: 291–300.
[36] J.L. Mott, M. Zafrullah. On Krull domains. Arch. Math., 1991, 56: 559–568.
[37] E. Kunz. Introduction to Commutative Algebra and Algebraic Geometry.Boston: Birkh¨auser, 1985.
[38] J.W. Brewer, W.J. Heinzer. Associated primes of principal ideals. DukeMath. J., 1974, 41: 1–7.
[39] S. Malik, J.L. Mott, M. Zafrullah. On t-invertibility. Comm. Algebra, 1988,16(1): 149–170.
[40] B.G. Kang. Pru¨fer v-multiplication domains and the ring R[X]Nv. J. Algebra,1989, 123: 151–170.
[41]王芳贵. GCD整环与UFD分式理想的刻划.科学通报, 1997, 42(5): 556.
[42] WANG Fanggui. On t-dimension over strong Mori domains. Acta Mathe-matica Sinica, English Series, 2006, 22(1): 131–138.
[43] N. Jacobson. Basic Algebra II. San Francisco: W.H. Freeman, 1980.
[44] E.G. Houston, T.G. Lucas, T.M. Viswanathan. Primary decomposition ofdivisorial ideals in Mori domains. J. Algebra, 1988, 117: 327–342.
[45] D.D. Anderson, B.G. Kang. Pseudo-Dedekind domains and divisorial idealsin R[X]T. J. Algebra, 1989, 122: 323–336.
[46] E. Houston, M. Zafrullah. On t-invertibility II. Comm. Algebra, 1989, 17(8):1955–1969.
[47] V. Barucci, D.F. Anderson, D.E. Dobbs. Coherent Mori domains and theprincipal ideal theorem. Comm. Algebra, 1987, 15(6): 1119–1156.
[48] M. Auslander, D.A. Buchsbaum. Homological dimension in Noetherian ringsII. Trans. Amer. Math. Soc., 1958, 88: 194–206.