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$$v_c$$ v c -Noetherian domains and Krull domains
Arabian Journal of Mathematics ( IF 0.9 ) Pub Date : 2021-02-23 , DOI: 10.1007/s40065-021-00318-0
Gyu Whan Chang

Let D be an integrally closed domain, \(\{V_{\alpha }\}\) be the set of t-linked valuation overrings of D, and \(v_c\) be the star operation on D defined by \(I^{v_c} = \bigcap _{\alpha } IV_{\alpha }\) for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a \(v_c\)-Noetherian domain if and only if D is a Krull domain, if and only if \(v_c = v\) and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then \(v_c = v\) if and only if D is a Dedekind domain.



中文翻译:

$$ v_c $$ vc -Noetherian域和Krull域

D为一个整体封闭域,\(\ {V _ {\ alpha} \} \)Dt链接估值环的集合,而\(v_c \)D\(I ^ {} V_C = \ bigcap _ {\阿尔法} IV _ {\阿尔法} \)对于所有非零分式理想d。在本文中,除其他外,我们证明D\( v_c \)- Noetherian域,当且仅当D是Krull域,并且当且仅当\(v_c = v \)并且每个素数t理想D是最大t-理想的。推论是,如果D是一维的,则当且仅当D是Dedekind域时,\(v_c = v \)

更新日期:2021-02-23
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