Throughout D is an integral domain with quotient field K and D ≠ K, and let * be a star operation on D. Let X be an indeterminate over D, and let K(X) be the quotient field of the polynomial ring K[X]. Set ${N}_{*}=\{f\in D[X]\mid c{(f)}^{*}=D\}$; then ${N}_{*}={N}_{{*}_{f}}={N}_{{*}_{w}}$ and $D{[X]}_{{N}_{*}}=\{\frac{g}{f}\mid g\in D[X]$ and $f\in {N}_{*}\}$ is an overring of D[X], which is also called the Nagata ring with respect to the star operation * and often denoted by Na(D, *). It is known that ${I}^{{*}_{w}}=ID{[X]}_{{N}_{*}}\cap K={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}I}{D}_{P}$ for all $I\in F(D)$ [3, Lemma 2.3] and *_{f}Max(D) = *_{w}Max(D) [1, Theorem 2.16].
Our first result is the extension [*] of *_{w} to D[X]. This is the star operation version of [4, Theorem 2.3], and hence the proofs are omitted.
Theorem 2.1.
Let X and Y be two indeterminates over D, and let
$$\begin{array}{l}\Delta =\{Q\in \text{Spec}(D[X])\mid Q\cap D=(0)\text{or}Q=(Q\cap D)[X]\text{and}{(Q\cap D)}^{{*}_{f}}\u228aD\}.\\ Set\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathcal{S}=D[X][Y]\setminus \left({\displaystyle \cup \{Q[Y]\mid Q\in \Delta \}}\right)and\text{\hspace{0.17em}}\text{\hspace{0.17em}}define\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}^{[*]}=A{[Y]}_{\mathcal{S}}\cap K(X)\text{\hspace{0.33em}}\text{\hspace{0.33em}}\text{\hspace{0.33em}}\text{forall}A\in F(D[X]).\end{array}$$
(1) The mapping [*] : F(D[X]) ⇒ F(D[X]), given by A ↦ A^{}[*]}, is a star operation on D[X] and [*]_{w} =[*].

(2) [*] = [*_{f}] =[*_{w}].

(3) (ID[X])^{}[*]} ∩ K = I^{}*_{w}} for all I ∈ F(D).

(4) (ID[X])^{}[*]} = I^{}*_{w}}D[X] for all I ∈ F(D).

(5) [*]Max$(D[X])=\{Q\mid Q\in $ Spec(D[X]) such that $Q\cap D=(0)$ and ${c}_{D}{(Q)}^{{*}_{f}}=D\}\cup \{P[X]\mid P\in {*}_{f}$Max(D)}.

(6) [v]=[t]=[w] is the woperation on D[X].
We next give some characterizations of the star operation [*] on D[X] which is introduced in Theorem 2.1.
Corollary 2.2.
If $A\in F(D[X])$, then

(1) ${A}^{[*]}=({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}D{[X]}_{P[X]})\cap AK[X]$

(2) ${A}^{[*]}=AD{[X]}_{{N}_{*}}\cap AK[X]$

(3) ${A}^{[*]}D{[X]}_{{N}_{*}}=AD{[X]}_{{N}_{*}}$ and ${A}^{[*]}D{[X]}_{P[X]}=AD{[X]}_{P[X]}$ for all $P\in {*}_{f}$Max(D).
Proof. (1) Note that ${A}^{{[*]}_{w}}={\displaystyle \underset{Q\in [*]\text{Max}(D[X])}{\cap}A}D{[X]}_{Q}$. Also note that $AK[X]=\cap \{AD{[X]}_{Q}\mid Q\in $ Spec(D[X]) and $Q\cap D=(0)\}$ and that if $Q\in $ Spec(D[X]) with $Q\cap D=(0)$ and ${c}_{D}{(Q)}^{{*}_{f}}\u228aD$, then $Q\subseteq P[X]$, and hence $D{[X]}_{P[X]}\subseteq D{[X]}_{Q}$ for some $P\in {*}_{f}Max(D)$. Thus by (1) and (5) of Theorem 2.1, we have ${A}^{[*]}=({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}D{[X]}_{P[X]})\cap (\cap \{Q\in Spec(D[X])\mid Q\cap D=(0)$ and ${c}_{D}{(Q)}^{{*}_{f}}=D\}=({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}D{[X]}_{P[X]})\cap AK[X]$.
(2) Recall from [12, Proposition 2.1] that $\left\{P{[X]}_{{N}_{*}}\rightP\in {*}_{f}Max(D)\}$ is the set of maximal ideals of $D{[X]}_{{N}_{*}}$; hence $AD{[X]}_{{N}_{*}}={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}{(AD{[X]}_{{N}_{*}})}_{P{[X]}_{{N}_{*}}}}$ = $\underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}D{[X]}_{P[X]$. Thus by (1), we have ${A}^{[*]}=AD{[X]}_{{N}_{*}}\cap AK[X]$.
(3) This is an immediate consequence of (1) and (2).
Next we introduce a new star operation {*} on D[X], which is an extension of *_{w} in the sense of ${(ID[X])}^{\{*\}}={I}^{{*}_{w}}[X]$ for all $I\in F(D)$ (Corollary 2.4(3)).
Theorem 2.3.
Let X and Y be two indeterminates over D. Let $N(*)=\{f\in D[Y]\mid c{(f)}^{*}=D\}$, and define
$${A}^{\{*\}}=A{[Y]}_{N(*)}\cap K(X)\text{\hspace{0.33em}}\text{\hspace{0.33em}}\text{\hspace{0.33em}}\text{forall}A\in F(D[X]).$$
(1) The mapping $\{*\}:F(D[X])\to F(D[X])$, given by $A\mapsto {A}^{\{*\}}$, is a star operation on D[X] and $\{*\}=\left\{{*}_{f}\right\}=\left\{{*}_{w}\right\}$.

(2) ${A}^{\{*\}}=\{u\in K(X)\mid uJ\subseteq A$ for some $J\in f(D)$ with ${J}^{*}=D\}$.

(3) ${A}^{\{*\}}={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}{D}_{P}[X]$.
Proof. (1) The property of $\{*\}=\left\{{*}_{f}\right\}=\left\{{*}_{w}\right\}$ is an immediate consequence of the fact that $N(*)=N({*}_{f})=N({*}_{w})$. Next, if ${(D[X])}^{\{*\}}=D[X]$, then the axioms for star operations are easily checked by the definition of {*}.
Clearly $D[X]\subseteq {(D[X])}^{\{*\}}$. For ${(D[X])}^{\{*\}}\subseteq D[X]$, let $f,g\in D[X]$, $u\in N(*)$ and $h\in D[X][Y]$ such that $\frac{g}{f}=\frac{h}{u}\in D[X]{[Y]}_{N(*)}\cap K(X)$. Then ug = fh and $g{c}_{D[X]}(u)=f{c}_{D[X]}(h)$. Note that ${c}_{D[X]}(u)={c}_{D}(u)[X]$, and hence
$${({c}_{D[X]}(u))}^{[*]}={({c}_{D}(u)[X])}^{[*]}={c}_{D}{(u)}^{{*}_{w}}[X]=D[X]$$
by Theorem 2.1(4). Hence $gD[X]=g{({c}_{D[X]}(u))}^{[*]}=f{({c}_{D[X]}(h))}^{[*]}\subseteq fD[X]$, and thus $\frac{g}{f}\in D[X]$. Therefore ${(D[X])}^{\{*\}}\subseteq D[X]$, and so ${(D[X])}^{\{*\}}=D[X]$.
(2) Let $B=\{u\in K(X)\mid uJ\subseteq A$ for some $J\in f(D)$ with ${J}^{*}=D\}$.
($\subseteq $) Let $u=\frac{g}{f}\in {A}^{\{*\}}=A{[Y]}_{N(*)}\cap K(X)$, where $g\in A[Y]$ and $f\in N(*)$. Then $uf=g$, and so $u{c}_{D}(f)\subseteq u{c}_{D}(f)[X]=u{c}_{D[X]}(f)={c}_{D[X]}(g)\subseteq A$ and ${c}_{D}{(f)}^{*}=D$. Thus $u\in B$.
($\supseteq $) If $u\in B$, there exists a $J\in f(D)$ with ${J}^{*}=D$ such that $uJ\subseteq A$. Choose $f\in D[Y]$ with ${c}_{D}(f)=J$. Then $f\in N(*)$, and hence $JD[X]{[Y]}_{N(*)}=D[X]{[Y]}_{N(*)}$. So $u\in uD[X]{[Y]}_{N(*)}\cap K(X)=uJD[X]{[Y]}_{N(*)}\cap K(X)\subseteq A{[Y]}_{N(*)}\cap K(X)={A}^{\{*\}}$. Thus $B\subseteq {A}^{\{*\}}$.
(3) Let B be as in the proof of (2). By (2), it suffices to show that
$$B={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}{D}_{P}[X].$$
($\subseteq $) $u\in B\Rightarrow uJ\subseteq A$ for some $J\in f(D)$ with ${J}^{*}=D\Rightarrow u\in u{D}_{P}[X]=uJ{D}_{P}[X]\subseteq A{D}_{P}[X]$ for all $P\in {*}_{f}Max(D)\Rightarrow u\in {\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}{D}_{P}[X]$.
($\supseteq $) For $v\in {\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}{D}_{P}[X]$, set $I=\{d\in D\mid dv\in A\}$. Then I is an ideal of D such that $I\u2288P$ for all $P\in {*}_{f}Max(D)$, and hence ${I}^{{*}_{f}}=D$. Since *_{f} is of finite character, there exists a $J\in f(D)$ such that $J\subseteq I$ and J^{* = D}. Hence $vJ\subseteq vI\subseteq A$, and thus $v\in B$.
In [5, Proposition 16], the authors studied the star operation {*} on D[X] defined in Theorem 2.3(3) in a more general setting of semistar operations (see Remark 2.6(1) for the definition of semistar operation). Hence, the properties (1)(4) and (6) of the next corollary were proved in [5, Propositions 16, 17 and Remark 19(c)].
Corollary 2.4.

(1) $\{*\}\lneqq [*]$.

(2) ${(ID[X])}^{\{*\}}\cap K={I}^{{*}_{w}}$ for all $I\in F(D)$.

(3) ${(ID[X])}^{\{*\}}={I}^{{*}_{w}}[X]={(ID[X])}^{[*]}$ for all $I\in F(D)$.

(4) $\{*\}={\{*\}}_{f}={\{*\}}_{w}$.

(5) ${A}^{\{*\}}{[Y]}_{N(*)}=A{[Y]}_{N(*)}$ and ${A}^{\{*\}}{D}_{P}[X]=A{D}_{P}[X]$ for all $P\in {*}_{f}Max(D)$.

(6) {d} is the doperation on D[X].
Proof. (1) If $A\in F(D[X])$, then by Corollary 2.2 and Theorem 2.3(2), we have ${A}^{\{*\}}={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}{D}_{P}[X]\subseteq ({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}A}D{[X]}_{P[X]})\cap AK[X]={A}^{[*]}$. Hence $\text{{}*\}\le [*]$. Also, if Q is a prime ideal of D[X] such that $Q\cap D\in {*}_{f}Max(D)$ and$(Q\cap D)[X]\u228aQ$, then ${Q}^{\{*\}}={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}Q}{D}_{P}[X]\subseteq Q{D}_{Q\cap D}[X]\cap D[X]=Q$, and hence ${Q}^{\{*\}}=Q$. But ${Q}^{[*]}=({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}Q}D{[X]}_{P[X]})\cap QK[X]=({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}D}{[X]}_{P[X]})\cap K[X]=D[X]$. Hence ${Q}^{\{*\}}\u228a{Q}^{[*]}$, and thus $\text{{}*\}\lneqq [*]$.
(2) ${(ID[X])}^{\{*\}}\cap K=(ID[X]{[Y]}_{N(*)}\cap K(X))\cap K=(ID{[Y]}_{N(*)})[X]\cap K=ID{[Y]}_{N(*)}\cap K={I}^{{*}_{w}}$ [3, Lemma 2.3].
(3) By (1) and (2), we have ${I}^{{*}_{w}}[X]\subseteq {(ID[X])}^{\{*\}}\subseteq {(ID[X])}^{[*]}$. Thus by Theorem 2.1(4), ${(ID[X])}^{\{*\}}={I}^{{*}_{w}}[X]={(ID[X])}^{[*]}$.
(4) Let $S=\{f\in D[X][Y]\mid {({c}_{D[X]}(f))}^{\{*\}}=D[X]\}$. If $g\in N(*)$, then ${c}_{D[X]}(g)={c}_{D}(g)[X]$. Hence ${({c}_{D[X]}(g))}^{\{*\}}={({c}_{D}(g)[X])}^{\{*\}}={c}_{D}{(g)}^{{*}_{w}}[X]=D[X]$ by (3), and so $g\in S$. Thus $N(*)\subseteq S$. So if $A\in F(D[X])$, then ${A}^{\{*\}}=A{[Y]}_{N(*)}\cap K(X)\subseteq A{[Y]}_{S}\cap K(X)={A}^{{\{*\}}_{w}}$ [3, Lemma 2.3]. Hence $\{*\}\le {\{*\}}_{w}$, and since ${\{*\}}_{w}\le {\{*\}}_{f}\le \{*\}$, we have $\{*\}={\{*\}}_{f}={\{*\}}_{w}$.
(5) This is an immediate consequence of the definition of {*} and Theorem 2.3(3).
(6) This follows directly from Theorem 2.3(2).
We say that D is a *Noetherian domain if D satisfies the ascending chain condition on integral *ideals. Hence a Noetherian domain is just the dNoetherian domain, while a Mori (resp., strong Mori) domain is a vNoetherian (resp., wNoetherian) domain. It is clear that if ${*}_{1}\le {*}_{2}$ are star operations on D, then *_{1}Noetherian domains are *_{2}Noetherian; hence Noetherian domains $\Rightarrow $ strong Mori domains $\Rightarrow $ Mori domains. Also, D is a *Noetherian domain if and only if each *ideal I of D is of finite type, i.e., I = J^{*} for some $J\in f(D)$; in particular, if D is *Noetherian, then *= *_{f}.
Corollary 2.5.
The following statements are equivalent.

(1) D is a *_{w}Noetherian domain.

(2) $D{[X]}_{{N}_{*}}$ is a Noetherian domain, where ${N}_{*}=\{f\in D[X]\mid {c}_{D}{(f)}^{*}=D\}$.

(3) D[X] is a [*]Noetherian domain.

(4) D[X] is a {*}Noetherian domain.
Proof. (1) $\iff $ (2) [3, Theorem 2.6 (1) $\iff $ (3)].
(2) $\Rightarrow $ (4) Let N(*) be as in Theorem 2.3. Then $(D{[Y]}_{N(*)})[X]=D[X]{[Y]}_{N(*)}$ is Noetherian by Hilbert basis theorem. Hence if A is a nonzero ideal of D[X], then $A{[Y]}_{N(*)}$ is finitely generated, i.e., $A{[Y]}_{N(*)}=({f}_{1},\dots ,{f}_{k})D[X]{[Y]}_{N(*)}$ for some ${f}_{1},\dots ,{f}_{k}\in A$. Thus ${A}^{\{*\}}=A{[Y]}_{N(*)}\cap K(X)=({f}_{1},\dots ,{f}_{k})D[X]{[Y]}_{N(*)}\cap K(X)={({f}_{1},\dots ,{f}_{k})}^{\{*\}}$.
(4) $\Rightarrow $ (3) This follows because $\{*\}\le [*]$ by Corollary 2.4(1).
(3) $\Rightarrow $ (1) Let ${I}_{1}\subseteq {I}_{2}\subseteq {I}_{3}\subseteq \cdots $ be a chain of *_{w}ideals of D. Then by Theorem 2.1(4), ${I}_{1}[X]\subseteq {I}_{2}[X]\subseteq {I}_{3}[X]\subseteq \cdots $ is a chain of [*]ideals of D[X]. Hence there exists an integer n such that ${I}_{n}[X]={I}_{k}[X]$ for all $k\ge n$, and thus ${I}_{n}={I}_{n}[X]\cap D={I}_{k}[X]\cap D={I}_{k}$ by Theorem 2.1(3).
Remark 2.6.
(1) Let $\overline{F}(D)$ be the set of nonzero Dsubmodules of K. Then $F(D)\subseteq \overline{F}(D)$. A semistar operation $\star $ on D is a mapping $E\mapsto {E}^{\star}$ of $\overline{F}(D)$ into $\overline{F}(D)$ that satisfies the following three conditions for all $0\ne x\in K$ and all $E,F\in \overline{F}(D)$:

(i) ${(xE)}^{\star}=x{E}^{\star}$

(ii) $E\subseteq {E}^{\star}$; $E\subseteq F$ implies ${E}^{\star}\subseteq {F}^{\star}$, and

(iii) ${({E}^{\star})}^{\star}={E}^{\star}$.
As in the star operation case, the ${\star}_{w}$operation is defined by ${E}^{{\star}_{w}}=\{x\in K\mid xJ\subseteq E$ for some $J\in f(D)$ with ${J}^{\star}={D}^{\star}\}$. It is clear that if ${D}^{\star}=D$, then the function $\star {}_{F(D)}:F(D)\to F(D)$, given by $I\mapsto {I}^{\star}$, is a star operation. Conversely, for any star operation *_{1} on D, define
$${E}^{{*}_{e}}=\left\{\begin{array}{l}{E}^{{*}_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}E\in F(D)\\ K,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}E\in \overline{F}(D)\setminus F(D).\end{array}\right.$$
Then the mapping ${*}_{e}:\overline{F}(D)\to \overline{F}(D),$ given by $E\mapsto {E}^{{*}_{e}}$, is a semistar operation and ${*}_{e}{}_{F(D)}={*}_{1}$.
(2) Let ⋆ be a semistar operation on D. A nonzero ideal I of D is called a quasi⋆ideal if $I={I}^{\star}\cap D$, while D is a ⋆Noetherian domain if D satisfies the ascending chain condition on quasi⋆ideals. Clearly, if ${D}^{\star}=D$, then I is a quasi⋆ideal if and only if $I={I}^{\star}$. Hence if ${D}^{\star}=D$, then D is ⋆Noetherian if and only if D is $\star {}_{F(D)}$Noetherian.
(3) Let $\mathcal{S}$ be the set of nonzero ideals B of D[X] such that $J[X]\subseteq B$ for some $J\in f(D)$ with ${J}^{\star}={D}^{\star}$, and set ${A}^{{\star}^{\prime}}=\{u\in K(X)\mid uB\subseteq A$ for some $B\in \mathcal{S}\}$ for all $A\in \overline{F}(D[X])$. Picozza proved that ${\star}^{\prime}$ is a semistar operation on D[X] (cf. [13, Propositions 3.1 and 3.2]) and that D is ${\star}_{w}$Nottherian if and only if D[X] is ${\star}^{\prime}$Noetherian [13, Theorem 3.6 (1) $\iff $ (2)].
(4) Suppose that ${D}^{\star}=D$, and set ${*}_{2}=\star {}_{F(D)}$. Then *_{2} is a star operation on D by (1). Let $A\in F(D)$. Note that $u\in {A}^{{\star}^{\prime}}\Rightarrow uB\subseteq A$ for some $B\in \mathcal{S}\Rightarrow uJ\subseteq A$ for some $J\in f(D)$ with ${J}^{{*}_{2}}=D\Rightarrow u\in {A}^{\left\{{*}_{2}\right\}}$. Conversely, $v\in {A}^{\left\{{*}_{2}\right\}}\Rightarrow vI\subseteq A$ for some $I\in f(D)$ with ${I}^{{*}_{2}}=D\Rightarrow vI[X]\subseteq A$ and $I[X]\in \mathcal{S}\Rightarrow v\in {A}^{{\star}^{\prime}}$. Hence ${A}^{\left\{{*}_{2}\right\}}={A}^{{\star}^{\prime}}$. Thus ${\star}^{\prime}{}_{F(D[X])}=\left\{{*}_{2}\right\}$, and so the equivalence of (1) and (4) of Corollary 2.5 is the star operation analog of [13, Theorem 3.6 (1) $\iff $ (2)].
(5) See [4, Corollary 2.5(2)] for the semistar Noetherian domain analog of the equivalence of (1) $\iff $ (3) of Corollary 2.5.
Let $\overline{D}$ be the integral closure of D. An element $x\in K$ is called *integral over D if there exists an $I\in f(D)$ such that $x{I}^{*}\subseteq {I}^{*}$. Let
$${D}^{[*]}=\{x\in K\mid x\text{\hspace{0.17em}}is\text{\hspace{0.17em}}*\text{\hspace{0.17em}}integral\text{\hspace{0.17em}}over\text{\hspace{0.17em}}D\}.$$
Then ${D}^{[*]}$, called the *integral closure of D, is an integrally closed domain and $\overline{D}\subseteq {D}^{[*]}$. Clearly if ${*}_{1}\le {*}_{2}$ are star operations on D, then ${D}^{[{*}_{1}]}\subseteq {D}^{[{*}_{2}]}$, and hence ${D}^{[{*}_{w}]}\subseteq {D}^{[{*}_{f}]}\subseteq {D}^{[*]}$. It is known that ${D}^{[{*}_{w}]}=\overline{D}{[X]}_{{N}_{*}}\cap K={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}\overline{{D}_{P}}}$ [3, Theorem 4.1] and ${(D[X])}^{[[v]]}={D}^{[w]}[X]$ [7, Proposition 1.7]. For more about *_{w}integral closure, see [3, 7].
Corollary 2.7.
${(D[X])}^{[[*]]}={(D[X])}^{[\{*\}]}={D}^{[{*}_{w}]}[X]$.
Proof. (i) Since $\text{{}*\}\lneqq [*]$ by Corollary 2.4(1), we have ${(D[X])}^{[\{*\}]}\subseteq {(D[X])}^{[[*]]}$.
(ii) (Proof of ${(D[X])}^{[[*]]}\subseteq {D}^{[{*}_{w}]}[X]$) Let $f\in {(D[X])}^{[[*]]}$. Then there exists a nonzero finitely generated ideal A of D[X] such that $f{A}^{[*]}\subseteq {A}^{[*]}$. Hence by Corollary 2.2(3), $fAD{[X]}_{P[X]}\subseteq AD{[X]}_{P[X]}$ for all $P\in {*}_{f}Max(D)$. Note that $AD{[X]}_{P[X]}$ is finitely generated; so $f\in \overline{D{[X]}_{P[X]}}\cap K[X]$. Note also that $\overline{D{[X]}_{P[X]}}\cap K[X]={\overline{{D}_{P}[X]}}_{{D}_{P}[X]\setminus P{D}_{P}[X]}\cap K[X]=\overline{{D}_{P}[X]}=\overline{{D}_{P}}[X]$; so $f\in \overline{{D}_{P}}[X]$. Hence each coefficient of f is contained in $({\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}\overline{{D}_{P}}}[X])\cap K={\displaystyle \underset{P\in {*}_{f}\text{Max}(D)}{\cap}\overline{{D}_{P}}}={D}^{[{*}_{w}]}$. Thus $f\in {D}^{[{*}_{w}]}[X]$.
(iii) (Proof of ${D}^{[{*}_{w}]}[X]\subseteq {(D[X])}^{[\{*\}]}$) Let $u\in {D}^{[{*}_{w}]}$. Then $u{J}^{{*}_{w}}\subseteq {J}^{{*}_{w}}$ for some $J\in f(D)$. Hence by Corollary 2.4(3), $u{(JD[X])}^{\{*\}}=u{J}^{{*}_{w}}D[X]\subseteq {J}^{{*}_{w}}D[X]={(JD[X])}^{\{*\}}$. Since JD[X] is finitely generated, we have $u\in {(D[X])}^{[\{*\}]}$. Hence ${D}^{[{*}_{w}]}\subseteq {(D[X])}^{[\{*\}]}$, and thus ${D}^{[{*}_{w}]}[X]\subseteq {(D[X])}^{[\{*\}]}$.