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 c_D(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)⊆ Invd(D)⊆ Inv*w(D)= Inv*f(D)⊆ Inv*(D)⊆ Invv(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.

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

Δ={Q∈Spec(D[X])∣Q∩D=(0) or Q=(Q∩D)[X] and (Q∩D)*f⊊D}.Set  S=D[X][Y]∖∪{Q[Y]∣Q∈Δ}and  define                                            A[*]=A[Y]S∩K(X)    for all A∈F(D[X]).

• (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

A{*}=A[Y]N(*)∩K(X)    for all A∈F(D[X]).

• (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])[{*}].

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, b_n=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.