Two generalizations of LCM-stable extensions
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let R ⊆ T be an extension of integral domains, X be anindeterminate over T, and R[X] and T[X] be polynomial rings. Then R ⊆ T is said to be LCM-stable if (aR∩bR)T = aT∩bT for all 0 ≠ a, b ∈ R. Let wA be the so-called w-operation on an integral domain A. In this paper,we introduce the notions of w(e)- and w-LCMstable extensions: (i) R ⊆ Tis w(e)-LCM-stable if ((aR ∩ bR)T)wT = aT ∩ bT for all 0 ≠ a, b ∈ R and (ii) R ⊆ T is w-LCM-stable if ((aR ∩ bR)T)wR = (aT ∩ bT)wR for all 0 ≠ a, b 2 R. We prove that LCM-stable extensions are both w(e)-LCM-stable and w-LCM-stable. We also generalize some results on LCM-stableextensions. Among other things, we show that if R is a Krull domain(resp., PvMD), then R ⊆ T is w(e)-LCM-stable (resp., w-LCM-stable) ifand only if R[X] ⊆ T[X] is w(e)-LCM-stable (resp., w-LCM-stable).
Abstract
Let R ⊆ T be an extension of integral domains, X be anindeterminate over T, and R[X] and T[X] be polynomial rings. Then R ⊆ T is said to be LCM-stable if (aR∩bR)T = aT∩bT for all 0 ≠ a, b ∈ R. Let wA be the so-called w-operation on an integral domain A. In this paper,we introduce the notions of w(e)- and w-LCMstable extensions: (i) R ⊆ Tis w(e)-LCM-stable if ((aR ∩ bR)T)wT = aT ∩ bT for all 0 ≠ a, b ∈ R and (ii) R ⊆ T is w-LCM-stable if ((aR ∩ bR)T)wR = (aT ∩ bT)wR for all 0 ≠ a, b 2 R. We prove that LCM-stable extensions are both w(e)-LCM-stable and w-LCM-stable. We also generalize some results on LCM-stableextensions. Among other things, we show that if R is a Krull domain(resp., PvMD), then R ⊆ T is w(e)-LCM-stable (resp., w-LCM-stable) ifand only if R[X] ⊆ T[X] is w(e)-LCM-stable (resp., w-LCM-stable).
참고문헌 (24)
- T. Akiba, "LCM-stableness, Q-stableness and flatness", Kobe J. Math, 2(1), pp.67~70 (1985)
- D. D. Anderson, "Two star-operations and their induced lattices", Comm. Algebra, 28(5), pp.2461~2475 (2000)
- D. D. Anderson, "t-linked extensions, the t-class group, and Nagata’s theorem", J. Pure Appl. Algebra, 86(2), pp.109~124 (1993)
- D. F. Anderson, "Overrings as intersections of localizations of an integral domain", ~
- N*, "J. Algebra", 2006, 216~233
- J. T. Condo, "LCM-stability of power series extensions characterizes Dedekind domains", Proc. Amer. Math. Soc, 123(8), pp.2333~2341 (1995)
- D. E. Dobbs, "On the criteria of D. D. Anderson for invertible and flat ideals", Canad. Math. Bull, 29(1), pp.25~32 (1986)
- D. E. Dobbs, "t-linked overrings and Pr¨ufer v-multiplication domains", Comm. Algebra, 17(11), pp.2835~2852 (1989)
- R. Gilmer, "An embedding theorem for HCF-rings", Proc. Cambridge Philos. Soc, 68, pp.583~587 (1970)
- R. Gilmer, "Finite element factorization in group rings, Ring theory", Lecture Notes in Pure and Appl. Math., 7, pp.47~61 (1974), Dekker
- R. Gilmer, "Multiplicative Ideal Theory", Queen’s Papers in Pure Appl. Math, 90 (1992), Queen’s University
- J. R. Hedstrom, "Some remarks on star-operations", J. Pure Appl. Algebra, 18(1), pp.37~44 (1980)
- E. G. Houston, "On t-invertibility. II", Comm. Algebra, 17(8), pp.1955~1969 (1989)
- B. G. Kang, "*-operations on integral domains", 박사, pp.1987
- Nv, "J. Algebra", 1989, 151~170
- D. J. Kwak, "On t-flat overrings", Chinese J. Math, 23(1), pp.17~24 (1995)
- A. Mimouni, "Integral domains in which each ideal is a w-ideal, Comm", Algebra, 33(5), pp.1345~1355 (2005)
- S. Oda, "Remarks on LCM-stableness and reflexiveness", Math. J. Toyama Univ, 17, pp.93~114 (1994)
- J. Sato, "The LCM-stability on polynomial extensions", Math. Rep. Toyama Univ, 10, pp.75~84 (1987)
- H. Uda, "LCM-stableness in ring extensions", Hiroshima Math. J, 13(2), pp.357~377 (1983)
- H. Uda, "G2-stableness and LCM-stableness", HiroshimaMath. J, 18(1), pp.47~52 (1988)
- F. Wang, "On w-modules over strong Mori domains", Comm. Al- gebra, 25(4), pp.1285~1306 (1997)
- Huayu Yin, "ω-MODULES OVER COMMUTATIVE RINGS", 대한수학회지, 48(1), pp.207~222 (2011), 대한수학회
- M. Zafrullah, "Putting t-invertibility to use, Non-Noetherian commutative ring theory", Math. Appl., 520, pp.429~457 (2000), Kluwer Acad. Publ.