*_Noetherian domains and the ring D[X]_N* , II
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Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D,N_* = <수식>, *w be the star operation on D defined by I*w = <수식>, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let A^g (resp., A^*g, A^([*]g)) be the global (resp., *-global, [*]-global) transform of a ring A. We show that D is a *w-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that D^(*g)[X]_(N*) = (D[X]_N* )^g = (D[X])[*]g; hence if D is a *w-Noetherian domain, then each ring between D[X]_(N*) and D^(*g)[X]_(N*)is a Noetherian domain. Let D = <수식> and ht P ≥ 2. We show that <수식> and study some properties of D and D^(*g).
Abstract
Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D,N_* = <수식>, *w be the star operation on D defined by I*w = <수식>, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let A^g (resp., A^*g, A^([*]g)) be the global (resp., *-global, [*]-global) transform of a ring A. We show that D is a *w-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that D^(*g)[X]_(N*) = (D[X]_N* )^g = (D[X])[*]g; hence if D is a *w-Noetherian domain, then each ring between D[X]_(N*) and D^(*g)[X]_(N*)is a Noetherian domain. Let D = <수식> and ht P ≥ 2. We show that <수식> and study some properties of D and D^(*g).
참고문헌 (20)
- D. D. Anderson, "Global transforms and Noetherian pairs", Hiroshima Math. J, 10(1), pp.69~74 (1980)
- D. D. Anderson, "Two star-operations and their induced lattices", Comm. Algebra, 28(5), pp.2461~2475 (2000)
- Nv, "J. Pure Appl. Algebra", 2005, 293~304
- N, "J. Algebra", 2006, 216~233
- G. W. Chang, "Prufer *-multiplication domains, Nagata rings, and Kronecker function rings", J. Algebra, 319(1), pp.309~319 (2008)
- Nv, "대한수학회지", 2008, 1405~1416, pp.대한수학회
- G. W. Chang, "Uppers to zero and semistar operations in polynomial rings", J. Algebra, 318(1), pp.484~493 (2007)
- G. W. Chang, "The w-integral closure of integral domains", J. Algebra, 295(1), pp.195~210 (2006)
- D. Dobbs, "t-linked overrings and Prufer v- multiplication domains", Comm. Algebra, 17(11), pp.2835~2852 (1989)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Dekker
- E. Houston, "Characterizations of *-multiplication domains", Canad. Math. Bull, 27(1), pp.48~52 (1984)
- Nv, "J. Algebra", 1989, 151~170
- I. Kaplansky, "Commutative rings, Revised Ed" (1974), Univ. of Chicago
- J. R. Matijevic, "Maximal ideal transforms of Noetherian rings", Proc. Amer. Math. Soc, pp.49~52 (1976)
- R. Matsuda, "On a question posed by Huckaba-Papick", Proc. Japan Acad. Ser. A Math. Sci, 59(1), pp.21~23 (1983)
- M. H. Park, "Group rings and semigroup rings over strong Mori domains", J. Pure Appl. Algebra, 163(3), pp.301~318 (2001)
- M. H. Park, "On overrings of strong Mori domains", J. Pure Appl. Algebra, 172(1), pp.79~85 (2002)
- A. R. Wadsworth, "Pairs of domains where all intermediate domains are Noetherian", Trans. Amer. Math. Soc, 195, pp.201~211 (1974)
- F. Wang, "On strong Mori domains", J. Pure Appl. Algebra, 135(2), pp.155~165 (1999)
- M. Zafrullah, "The v-operation and intersections of quotient rings of integral domains", Comm. Algebra, 13(8), pp.1699~1712 (1985)