Article Search
eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(1): 23-32

Published online March 31, 2021

### Two Extensions of a Star Operation on D to the Polynomial Ring D[X]

Gyu Whan Chang, Hwankoo Kim*

Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea
e-mail : whan@inu.ac.kr

Division of Computer and Information Engineering, Hoseo University, Asan 31499, Republic of Korea
e-mail : hkkim@hoseo.edu

Received: February 22, 2020; Revised: March 18, 2020; Accepted: May 30, 2020

Let D be an integral domain with quotient field K, X an indeterminate over D, * a star operation on D, and $Cl*(D)$ be the *-class group of D. The $*w$-operation on $D$ is a star operation defined by $I*w={x∈K∣xJ⊆I$ for a nonzero finitely generated ideal J of D with $J*=D}$. In this paper, we study two star operations ${*}$ and [*] on D[X] defined by $A{*}=∩ P∈*w-Max(D)ADP [X]$ and $A[*]=(∩ P∈*w-Max(D)AD[X]P[X] )∩AK[X]$. Among other things, we show that $Cl*(D)≅Cl[*](D[X])$ if and only if D is integrally closed.

Keywords: star operation, extension of a star operation to the polynomial ring, t-class group, integrally closed, *-Noetherian domain.

Let D be an integral domain with quotient field K, and assume that D is not a field, i.e., $D≠K$. Let X be an indeterminate over D. For a polynomial $f∈K[X]$, let cD(f) (or simply c(f)) be the fractional ideal of D generated by the coefficients of f. For a fractional ideal A of D[X], let $cD(A)=∑{cD(f)∣f∈A}$; hence $cD(A)$ is a fractional ideal of D. Let $f(D)$ (resp., $f(D)$) be the set of nonzero (resp., nonzero finitely generated) fractional ideals of D; so $f(D)⊆F(D)$.

Let * be a star operation on D (see the next paragraph for definitions related to star operations). We say that a star operation ∑ on D[X] is an extension of * if $(ID[X])⋆∩K=I*$ for all $I∈F(D)$. In this paper, we study two extensions [*] and {*} of * to D[X]; the star operation [*] (resp., {*}) on D[X], first studied in [4] (resp., [5]), is defined by $A[*]=(∩ P∈*w-Max(D)AD[X]P[X])∩AK[X]$ (resp., $A{*}=∩ P∈*w-Max(D)ADP[X])$ for all $A∈F(D[X])$; hence $A[w]=Aw$ and $A[d]=A$. We first study some properties of [*] and {*}. Then, as a corollary, we show that D is a *w-Noetherian domain if and only if D[X] is a {*}-Noetherian domain, if and only if D[X] is a [*]-Noetherian domain. Let $(D[X])[[*]]$ and $(D[X])[{*}]$ be the [*]- and {*}-integral closures of D[X] and let $D[*w]$ be the *w-integral closure of D. We prove that $(D[X])[[*]]=(D[X])[{*}]=D[*w][X]$. Finally we prove that $Cl*(D)≅Cl[*](D[X])$ if and only if D is integrally closed. This is a generalization of the following well-known result: $Clt(D)≅Clt(D[X])$ if and only if D is integrally closed [8, Theorem 3.6].

A star operation on D is a mapping $I↦I*$ of $F(D)$ into $F(D)$ that satisfies the following three conditions for all $0≠a∈K$ and all $I,J∈F(D)$: (i) $(aD)*=aD$ and $(aI)*=aI*$, (ii) $I⊆I*$; $I⊆J$ implies $I*⊆J*$, and (iii) $(I*)*=I*$. Given a star operation * on D, we can construct two new star operations *f and *w on D as follows: $I*f=∪{J*∣J⊆I$ and $J∈f(D)}$ and $I*w={x∈K∣xJ⊆I$ for some $J∈f(D)$ with $J*=D}$ for all $I∈F(D)$. The v-operation is a star operation defined by $Iv=(I−1)−1$, where $I−1={x∈K∣xI⊆D}$. The t-operation is defined by $t=vf$ and the w-operation is defined by $w=vw$. The d-operation is just the identity function on $F(D)$; so $d=df=dw$.

Let * be a star operation on D. Clearly, $I*=I*f$ for all $I∈f(D)$ and $(*f)f=*f$. We say that * is of finite character if * = *f, i.e., if $I*=I*f$ for all $I∈f(D)$; hence *f is of finite character. An $I∈f(D)$ is called a *-ideal if I = I*. Let *-Max(D) denote the set of *-ideals maximal among proper integral *-ideals of D. A *-ideal in *-Max(D) is called a maximal *-ideal. It is well known that $*f−Max(D)≠∅$, $D=∩ P∈*f-Max(D)DP$, each integral *f-ideal is contained in a maximal *f-ideal and $(*w)f=*w=(*f)w$. Let *1 and *2 be two star operations on D. We mean by $*1≤*2$ that $I*1⊆I*2$ for all $I∈F(D)$. Clearly, $*w≤*f≤*$. Also, if $*1≤*2$, then $(*1)w≤(*2)w$ and $(*1)f≤(*2)f$. It is well known that $d≤*≤v$, and hence $d≤*w≤w≤t≤v$ and $d≤*f≤t$.

An $I∈F(D)$ is called *-invertible if $(II−1)*=D$. Let Inv*(D) be the set of *-invertible *-ideals of D. Then Inv*(D) forms an abelian group under the usual *-multiplication $I×J:=(IJ)*$. Let Prin(D) be the subgroup of Inv*(D) consisting of nonzero principal fractional ideals of D. Hence, Prin$(D)⊆$ Inv$d(D)⊆$ Inv$*w(D)=$ Inv$*f(D)⊆$ Inv$*(D)⊆$ Inv$v(D)$. The *-class group of D is the factor group $Cl*(D)=$ Inv*(D)/Prin(D). For a *-invertible *-ideal I of D, let [I] denote the ideal class of $Cl*(D)$ containing I. Hence, [I]= [J] for *-invertible *-ideals I,J if and only if I = aJ for some $a∈K$. It is clear that if $*1≤*2$ are star operations on D, then Inv$*1(D)⊆$ Inv*2(D), and thus $Cl*1(D)⊆Cl*2(D)$.

The reader can be referred to [9, Section 32] for basic properties of star operations.

### 2. Two Extensions [*] and {*} of a Star Operation * on D

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 $N*={f∈D[X]∣c(f)*=D}$; then $N*=N*f=N*w$ and $D[X]N*={gf∣g∈D[X]$ and $f∈N*}$ 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 $I*w=ID[X]N*∩K=∩ P∈ *f Max(D)IDP$ for all $I∈F(D)$ [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

• (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$(D[X])={Q∣Q∈$ Spec(D[X]) such that $Q∩D=(0)$ and $cD(Q)*f=D}∪{P[X]∣P∈*f$-Max(D)}.

• (6) [v]=[t]=[w] is the w-operation 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 $A∈F(D[X])$, then

• (1) $A[*]=(∩ P∈*f-Max(D)AD[X]P[X])∩AK[X]$

• (2) $A[*]=AD[X]N*∩AK[X]$

• (3) $A[*]D[X]N*=AD[X]N*$ and $A[*]D[X]P[X]=AD[X]P[X]$ for all $P∈*f$-Max(D).

Proof. (1) Note that $A[*]w=∩ Q∈[*]-Max(D[X])AD[X]Q$. Also note that $AK[X]=∩{AD[X]Q∣Q∈$ Spec(D[X]) and $Q∩D=(0)}$ and that if $Q∈$ Spec(D[X]) with $Q∩D=(0)$ and $cD(Q)*f⊊D$, then $Q⊆P[X]$, and hence $D[X]P[X]⊆D[X]Q$ for some $P∈*f−Max(D)$. Thus by (1) and (5) of Theorem 2.1, we have $A[*]=(∩ P∈*f-Max(D)AD[X]P[X])∩(∩{Q∈Spec(D[X])∣Q∩D=(0)$ and $cD(Q)*f=D}=(∩ P∈ * f-Max(D)AD[X]P[X])∩AK[X]$.

(2) Recall from [12, Proposition 2.1] that ${P[X]N*|P∈*f−Max(D)}$ is the set of maximal ideals of $D[X]N*$; hence $AD[X]N*=∩ P∈ *f -Max(D)(AD[X]N* )P[X]N*$ = $∩ P∈*f-Max(D)AD[X]P[X]$. Thus by (1), we have $A[*]=AD[X]N*∩AK[X]$.

(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 $I∈F(D)$ (Corollary 2.4(3)).

### Theorem 2.3.

Let X and Y be two indeterminates over D. Let $N(*)={f∈D[Y]∣c(f)*=D}$, and define

• (1) The mapping ${*}:F(D[X])→F(D[X])$, given by $A↦A{*}$, is a star operation on D[X] and ${*}={*f}={*w}$.

• (2) $A{*}={u∈K(X)∣uJ⊆A$ for some $J∈f(D)$ with $J*=D}$.

• (3) $A{*}=∩ P∈*f-Max(D)ADP[X]$.

Proof. (1) The property of ${*}={*f}={*w}$ 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 $D[X]⊆(D[X]){*}$. For $(D[X]){*}⊆D[X]$, let $f,g∈D[X]$, $u∈N(*)$ and $h∈D[X][Y]$ such that $gf=hu∈D[X][Y]N(*)∩K(X)$. Then ug = fh and $gcD[X](u)=fcD[X](h)$. Note that $cD[X](u)=cD(u)[X]$, and hence

$(cD[X](u))[*]=(cD(u)[X])[*]=cD(u)*w[X]=D[X]$

by Theorem 2.1(4). Hence $gD[X]=g(cD[X](u))[*]=f(cD[X](h))[*]⊆fD[X]$, and thus $gf∈D[X]$. Therefore $(D[X]){*}⊆D[X]$, and so $(D[X]){*}=D[X]$.

(2) Let $B={u∈K(X)∣uJ⊆A$ for some $J∈f(D)$ with $J*=D}$.

($⊆$) Let $u=gf∈A{*}=A[Y]N(*)∩K(X)$, where $g∈A[Y]$ and $f∈N(*)$. Then $uf=g$, and so $ucD(f)⊆ucD(f)[X]=ucD[X](f)=cD[X](g)⊆A$ and $cD(f)*=D$. Thus $u∈B$.

($⊇$) If $u∈B$, there exists a $J∈f(D)$ with $J*=D$ such that $uJ⊆A$. Choose $f∈D[Y]$ with $cD(f)=J$. Then $f∈N(*)$, and hence $JD[X][Y]N(*)=D[X][Y]N(*)$. So $u∈uD[X][Y]N(*)∩K(X)=uJD[X][Y]N(*)∩K(X)⊆A[Y]N(*)∩K(X)=A{*}$. Thus $B⊆A{*}$.

(3) Let B be as in the proof of (2). By (2), it suffices to show that

$B=∩ P∈*f-Max(D)ADP[X].$

($⊆$) $u∈B⇒uJ⊆A$ for some $J∈f(D)$ with $J*=D⇒u∈uDP[X]=uJDP[X]⊆ADP[X]$ for all $P∈*f−Max(D)⇒u∈∩ P∈ * f-Max(D)ADP[X]$.

($⊇$) For $v∈∩ P∈*f-Max(D)ADP[X]$, set $I={d∈D∣dv∈A}$. Then I is an ideal of D such that $I⊈P$ for all $P∈*f−Max(D)$, and hence $I*f=D$. Since *f is of finite character, there exists a $J∈f(D)$ such that $J⊆I$ and J* = D. Hence $vJ⊆vI⊆A$, and thus $v∈B$.

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) ${*}≨[*]$.

• (2) $(ID[X]){*}∩K=I*w$ for all $I∈F(D)$.

• (3) $(ID[X]){*}=I*w[X]=(ID[X])[*]$ for all $I∈F(D)$.

• (4) ${*}={*}f={*}w$.

• (5) $A{*}[Y]N(*)=A[Y]N(*)$ and $A{*}DP[X]=ADP[X]$ for all $P∈*f−Max(D)$.

• (6) {d} is the d-operation on D[X].

Proof. (1) If $A∈F(D[X])$, then by Corollary 2.2 and Theorem 2.3(2), we have $A{*}=∩ P∈*f-Max(D)ADP[X]⊆(∩ P∈*f-Max(D)AD[X]P[X])∩AK[X]=A[*]$. Hence ${*}≤[*]$. Also, if Q is a prime ideal of D[X] such that $Q∩D∈*f−Max(D)$ and$(Q∩D)[X]⊊Q$, then $Q{*}=∩ P∈*f-Max(D)QDP[X]⊆QDQ∩D[X]∩D[X]=Q$, and hence $Q{*}=Q$. But $Q[*]=(∩ P∈*f-Max(D)QD[X]P[X])∩QK[X]=(∩ P∈*f-Max(D)D[X]P[X])∩K[X]=D[X]$. Hence $Q{*}⊊Q[*]$, and thus ${*}≨[*]$.

(2) $(ID[X]){*}∩K=(ID[X][Y]N(*)∩K(X))∩K=(ID[Y]N(*))[X]∩K=ID[Y]N(*)∩K=I*w$ [3, Lemma 2.3].

(3) By (1) and (2), we have $I*w[X]⊆(ID[X]){*}⊆(ID[X])[*]$. Thus by Theorem 2.1(4), $(ID[X]){*}=I*w[X]=(ID[X])[*]$.

(4) Let $S={f∈D[X][Y]∣(cD[X](f)){*}=D[X]}$. If $g∈N(*)$, then $cD[X](g)=cD(g)[X]$. Hence $(cD[X](g)){*}=(cD(g)[X]){*}=cD(g)*w[X]=D[X]$ by (3), and so $g∈S$. Thus $N(*)⊆S$. So if $A∈F(D[X])$, then $A{*}=A[Y]N(*)∩K(X)⊆A[Y]S∩K(X)=A{*}w$ [3, Lemma 2.3]. Hence ${*}≤{*}w$, and since ${*}w≤{*}f≤{*}$, 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 d-Noetherian domain, while a Mori (resp., strong Mori) domain is a v-Noetherian (resp., w-Noetherian) domain. It is clear that if $*1≤*2$ are star operations on D, then *1-Noetherian domains are *2-Noetherian; hence Noetherian domains $⇒$ strong Mori domains $⇒$ 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 $J∈f(D)$; 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 $N*={f∈D[X]∣cD(f)*=D}$.

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

• (4) D[X] is a {*}-Noetherian domain.

Proof. (1) $⇔$ (2) [3, Theorem 2.6 (1) $⇔$ (3)].

(2) $⇒$ (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., $A[Y]N(*)=(f1,…,fk)D[X][Y]N(*)$ for some $f1,…,fk∈A$. Thus $A{*}=A[Y]N(*)∩K(X)=(f1,…,fk)D[X][Y]N(*)∩K(X)=(f1,…,fk){*}$.

(4) $⇒$ (3) This follows because ${*}≤[*]$ by Corollary 2.4(1).

(3) $⇒$ (1) Let $I1⊆I2⊆I3⊆⋯$ be a chain of *w-ideals of D. Then by Theorem 2.1(4), $I1[X]⊆I2[X]⊆I3[X]⊆⋯$ is a chain of [*]-ideals of D[X]. Hence there exists an integer n such that $In[X]=Ik[X]$ for all $k≥n$, and thus $In=In[X]∩D=Ik[X]∩D=Ik$ by Theorem 2.1(3).

### Remark 2.6.

(1) Let $F¯(D)$ be the set of nonzero D-submodules of K. Then $F(D)⊆F¯(D)$. A semistar operation $⋆$ on D is a mapping $E↦E⋆$ of $F¯(D)$ into $F¯(D)$ that satisfies the following three conditions for all $0≠x∈K$ and all $E,F∈F¯(D)$:

• (i) $(xE)⋆=xE⋆$

• (ii) $E⊆E⋆$; $E⊆F$ implies $E⋆⊆F⋆$, and

• (iii) $(E⋆)⋆=E⋆$.

As in the star operation case, the $⋆w$-operation is defined by $E⋆w={x∈K∣xJ⊆E$ for some $J∈f(D)$ with $J⋆=D⋆}$. It is clear that if $D⋆=D$, then the function $⋆|F(D):F(D)→F(D)$, given by $I↦I⋆$, is a star operation. Conversely, for any star operation *1 on D, define

$E*e=E*1, if E∈F(D)K, if E∈F¯(D)∖F(D).$

Then the mapping $*e:F¯(D)→F¯(D),$ given by $E↦E*e$, 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 $I=I⋆∩D$, while D is a ⋆-Noetherian domain if D satisfies the ascending chain condition on quasi-⋆-ideals. Clearly, if $D⋆=D$, then I is a quasi-⋆-ideal if and only if $I=I⋆$. Hence if $D⋆=D$, then D is ⋆-Noetherian if and only if D is $⋆|F(D)$-Noetherian.

(3) Let $S$ be the set of nonzero ideals B of D[X] such that $J[X]⊆B$ for some $J∈f(D)$ with $J⋆=D⋆$, and set $A ⋆′={u∈K(X)∣uB⊆A$ for some $B∈S}$ for all $A∈F¯(D[X])$. Picozza proved that $⋆′$ is a semistar operation on D[X] (cf. [13, Propositions 3.1 and 3.2]) and that D is $⋆w$-Nottherian if and only if D[X] is $⋆′$-Noetherian [13, Theorem 3.6 (1) $⇔$ (2)].

(4) Suppose that $D⋆=D$, and set $*2=⋆|F(D)$. Then *2 is a star operation on D by (1). Let $A∈F(D)$. Note that $u∈A⋆′⇒uB⊆A$ for some $B∈S⇒uJ⊆A$ for some $J∈f(D)$ with $J*2=D⇒u∈A{*2}$. Conversely, $v∈A{*2}⇒vI⊆A$ for some $I∈f(D)$ with $I*2=D⇒vI[X]⊆A$ and $I[X]∈S⇒v∈A⋆′$. Hence $A{*2}=A ⋆′$. Thus $⋆′|F(D[X])={*2}$, and so the equivalence of (1) and (4) of Corollary 2.5 is the star operation analog of [13, Theorem 3.6 (1) $⇔$ (2)].

(5) See [4, Corollary 2.5(2)] for the semistar Noetherian domain analog of the equivalence of (1) $⇔$ (3) of Corollary 2.5.

Let $D¯$ be the integral closure of D. An element $x∈K$ is called *-integral over D if there exists an $I∈f(D)$ such that $xI*⊆I*$. Let

$D[*]={x∈K∣x is * −integral over D}.$

Then $D[*]$, called the *-integral closure of D, is an integrally closed domain and $D¯⊆D[*]$. Clearly if $*1≤*2$ are star operations on D, then $D[*1]⊆D[*2]$, and hence $D[*w]⊆D[*f]⊆D[*]$. It is known that $D[*w]=D¯[X]N*∩K=∩ P∈ *f -Max(D)DP¯$ [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 ${*}≨[*]$ by Corollary 2.4(1), we have $(D[X])[{*}]⊆(D[X])[[*]]$.

(ii) (Proof of $(D[X])[[*]]⊆D[*w][X]$) Let $f∈(D[X])[[*]]$. Then there exists a nonzero finitely generated ideal A of D[X] such that $fA[*]⊆A[*]$. Hence by Corollary 2.2(3), $fAD[X]P[X]⊆AD[X]P[X]$ for all $P∈*f−Max(D)$. Note that $AD[X]P[X]$ is finitely generated; so $f∈D[X]P[X]¯∩K[X]$. Note also that $D[X]P[X]¯∩K[X]=DP[X]¯DP[X]∖PDP[X]∩K[X]=DP[X]¯=DP¯[X]$; so $f∈DP¯[X]$. Hence each coefficient of f is contained in $(∩ P∈*f-Max(D)DP ¯[X])∩K=∩ P∈*f-Max(D)DP ¯=D[*w]$. Thus $f∈D[*w][X]$.

(iii) (Proof of $D[*w][X]⊆(D[X])[{*}]$) Let $u∈D[*w]$. Then $uJ*w⊆J*w$ for some $J∈f(D)$. Hence by Corollary 2.4(3), $u(JD[X]){*}=uJ*wD[X]⊆J*wD[X]=(JD[X]){*}$. Since JD[X] is finitely generated, we have $u∈(D[X])[{*}]$. Hence $D[*w]⊆(D[X])[{*}]$, and thus $D[*w][X]⊆(D[X])[{*}]$.

### 3. The Star Class Group of Polynomial Rings

Let D be an integral domain with quotient field K, X be an indeterminate over D, * be a star operation of finte type on D, and [*] and {*} be the star operations of finite type on D[X] as in Theorems 2.1 and 2.3.

### Lemma 3.1.

$Cl*(D)⊆Cl{*}(D[X])⊆Cl[*](D[X])$.

Proof. (Proof of $Cl*(D)⊆Cl{*}(D[X])$) Let $I∈Inv*(D)$. Then $(II−1)*w=D$, and hence $(ID[X](ID[X])−1){*}=(ID[X](I−1D[X])){*}=((II−1)D[X]){*}=(II−1)*wD[X]=D[X]$ (cf., Corollary 2.4(3) for the third equality). Thus the map $φ:Cl*(D)→Cl{*}(D[X])$, given by $[I]↦[ID[X]]$, is well-defined.

Next, let $I∈Inv*(D)$ such that $(ID[X]){*}$ is principal. Since $(ID[X]){*}=I*wD[X]$, we have $I*wD[X]=fD[X]$ for some $f∈I*wD[X]$. But, since $I*wD[X]∩K=I*w≠(0)$, we have $f∈K$, and hence $I*w=I*wD[X]∩K=fD[X]∩K=(fD)*w=fD$. Thus, $φ$ is injective. Therefore, $Cl*(D)⊆Cl{*}(D[X])$.

(Proof of $Cl{*}(D[X])⊆Cl[*](D[X])$) This follows because ${*}≤[*]$ by Corollary 2.4.

### Theorem 3.2.

$Cl*(D)=Cl[*](D[X])$ if and only if D is integrally closed.

Proof. $(⇒)$ For $α=ab∈K$ such that $a,b∈D$ and α is integral over D, let f = bX -a and $Q=fK[X]∩D[X]$. Since α is integral over D, there exists a monic polynomial $g∈D[X]$ such that $g(α)=0$. Then $g∈Q$, and hence $c(Q)=D$, or $c(Q)t=D$. Hence Q is a maximal t-ideal, and so Q is t-invertible [11, Theorem 1.4]. Since Q is a t-ideal, we have $Q⊊QQ−1$. Note that $c(Q)*=D$; hence Q is a maximal [*]-ideal of D[X] by Theorem 2.1(5), and thus $(QQ−1)[*]=D[X]$.

Next, since $Cl*(D)=Cl[*](D[X])$, there exists an integral ideal I of D and $u∈K(X)$ such that $ID[X]=Qu$. For $0≠c∈I$, c = qu for some $q∈Q$; so $u=cq$. Since $f∈Q$, we have $fcq∈ID[X]$. Also, since deg(f)=1, we have deg$(q)≤1$. If deg(q)=0, i.e., $q∈K$, then $I=ID[X]∩D=Qu∩D=(0)$, a contradiction. Thus deg(q) = 1. Note that $g∈Q$ and $gcq∈ID[X]$. Hence $gcq=∑ i=0nciXi∈ID[X]$. Then $gc=∑ i=0n(ciq)Xi$. Since each $qci∈qID[X]=Qc$, there exist some $ai,bi∈D$ such that $ciq=(biX−ai)c$ and $biX−ai∈Q$. Hence $gc=∑ i=0nXi(biX−ai)c$, and so $g=∑ i=0nXi(biX−ai)$. Note that $biα−ai=0$ for each i, and so $α=aibi$. Since g is monic, bn=1, and thus $α=anbn=an∈D$.

$(⇐)$ By Lemma 3.1, it suffices to show that $φ$ is surjective. Let A be a [*]-invertible integral [*]-ideal of D[X]. Then A is a t-invertible t-ideal [6, Lemma 2.1], and hence $A=fID[X]$ for some $f∈D[X]$ and a t-invertible t-ideal I of D [8, Lemma 3.3]. Note that $A=A[*]=(fID[X])[*]=fI*wD[X]$ by Theorem 2.1(4). Hence I is a *-invertible *-ideal, and $[A]=[ID[X]]=φ([I])$. Thus $φ$ is surjective.

### Corollary 3.3.

([8, Theorem 3.6]) $Clt(D)≅Clt(D[X])$ if and only if D is integrally closed.

Proof. This follows from Theorem 3.2 because $Clt(D)=Clw(D)$ and $Cl[w](D[X])=Clw(D[X])=Clt(D[X])$.

An integral domain D with quotient field K is said to be seminormal if $α∈K$ with $α2,α3∈D$ implies that $α∈D$. Then the following theorem is well known: $Pic(D)≅Pic(D[X])$ if and only if D is seminormal [10, Theorem 1.6]. The proof of the next result is essentially the same as that in [2, p. 209]. For the sake of completeness we give its proof.

### Proposition 3.4.

If the map $φ$ in Lemma 3.1 is an isomorphism, then D is seminormal.

Proof. Let K be the quotient field of D. Assume that D is not seminormal. Then there exists $α∈K$ such that $α∈D$, but $α2,α3∈D$. Consider the fractional ideals $I:=(α2,1+αX)$ and $J:=(α2,1−αX)$ of D[X]. Then $IJ=(α4,α2+α3X,α2−α3X,1−α2X2)⊆D[X]$. Now the equality $X4α4+(1+α2X2)(1−α2X2)=1$ implies that $IJ=D[X]$, and so I and J are invertible, with $J=I−1$. Hence I is a {*}-invertible {*}-ideal of D[X]. As claimed in [2, p. 209], I is not extended from D, which implies that $φ$ is not an isomorphism.

We do not know whether the converse of Proposition 3.4 holds.

The authors would like to thank the reviewer for comments and suggestions. This work was supported by the Incheon National University Research Grant in 2019.

1. D. D. Anderson and S. J. Cook. Two star operations and their induced lattices, Comm. Algebra 28(2000), 2461-2475.
2. J. W. Brewer and D. L. Costa. Seminormality and projective modules over polynomial rings, J. Algebra 58(1979), 208-216.
3. G. W. Chang. *-Noetherian domains and the ring D[X]N*, J. Algebra 297(2006), 216-233.
4. G. W. Chang and M. Fontana. Upper to zero and semistar operations in polynomial rings, J. Algebra 318(2007), 484-493.
5. G. W. Chang and M. Fontana. An overring-theoretic approach to polynomial extensions of star and semistar operations, Comm. Algebra 39(2011), 1956-1978.
6. G. W. Chang and J. Park. Star-invertible ideals of integral domains, Boll. Unione. Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6(2003), 141-150.
7. G. W. Chang and M. Zafrullah. The w-integral closure of integral domains, J. Algebra 259(2006), 195-210.
8. S. Gabelli. On divisorial ideals in polynomial rings over Mori domains, Comm. Algebra 15(1987), 2349-2370.
9. R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
10. R. Gilmer and R. Heitmann. On Pic(R[X]) for R seminormal, J. Pure Appl. Algebra 16(1980), 251-257.
11. E. Houston and M. Zafrullah. On t-invertibility II, Comm. Algebra 17(1989), 1955-1969.
12. B. G. Kang. Prüfer v-multiplication domains and the ring R[X]Nv, J. Algebra 123(1989), 151-170.
13. G. Picozza. A note on semistar Noetherian domains, Houston J. Math. 33(2007), 415-432.