*-NOETHERIAN DOMAINS AND THE RING D[X]N*, II

Title & Authors
*-NOETHERIAN DOMAINS AND THE RING D[X]N*, II
Chang, Gyu-Whan;

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, $\small{N_*}$={f $\small{\in}$ D[X]|c(f)$\small{^*}$= D}, $\small{*_w}$ be the star operation on D defined by $\small{I^{*_w}}$ = ID[X]$\small{{_N}_*}$ $\small{\cap}$ K, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let $\small{A^g}$ (resp., $\small{A^{[*]g}}$, $\small{A^{[*]g}}$) be the global (resp.,*-global, [*]-global) transform of a ring A. We show that D is a $\small{*_w}$-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that $\small{D^{*g}}$[X]$\small{{_N}_*}$ = (D[X]$\small{{_N}_*}$)$\small{^g}$ = (D[X])$\small{^{[*]g}}$; hence if D is a $\small{*_w}$-Noetherian domain, then each ring between D[X]$\small{{_N}_*}$ and $\small{D^{*g}}$[X]$\small{{_N}_*}$ is a Noetherian domain. Let $\small{\tilde{D}}$ = $\small{\cap}${$\small{D_P}$|P $\small{\in}$ $\small{*_w}$-Max(D) and htP $\small{\geq}$2}. We show that $\small{D\;\subseteq\;\tilde{D}\;\subseteq\;D^{*g}}$ and study some properties of $\small{\tilde{D}}$ and $\small{D^{*g}}$.
Keywords
star operation;[*]-operation on D[X];*-global transform;$\small{*_w}$-Noetherian domain;D[X]$\small{{_N}_*}$;SM domain pair;
Language
English
Cited by
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