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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).
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참고문헌 (24)
참고문헌 신청
A. Mimouni / 2005 / Integral domains in which each ideal is a w-ideal, Comm / Algebra 33 (5) : 1345 ~ 1355
B. G. Kang / 1987 / *-operations on integral domains / 박사 / Univ. Iowa
B. G. Kang / 1989 / Pr¨ufer v-multiplication domains and the ring R[X]Nv / J. Algebra 123 (1) : 151 ~ 170
D. D. Anderson / 1993 / t-linked extensions, the t-class group, and Nagata’s theorem / J. Pure Appl. Algebra 86 (2) : 109 ~ 124
D. D. Anderson / 2000 / Two star-operations and their induced lattices / Comm. Algebra 28 (5) : 2461 ~ 2475
B. G. Kang / 1987 / *-operations on integral domains / 박사 / Univ. Iowa
B. G. Kang / 1989 / Pr¨ufer v-multiplication domains and the ring R[X]Nv / J. Algebra 123 (1) : 151 ~ 170
D. D. Anderson / 1993 / t-linked extensions, the t-class group, and Nagata’s theorem / J. Pure Appl. Algebra 86 (2) : 109 ~ 124
D. D. Anderson / 2000 / Two star-operations and their induced lattices / Comm. Algebra 28 (5) : 2461 ~ 2475
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