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

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