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