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

pISSN 1225-6951 eISSN 0454-8124 c

Kyungpook Mathematical Journal

Two Extensions of a Star Operation on D to the Polynomial

Ring D[X]

Gyu Whan Chang

Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea

e-mail : whan@inu.ac.kr Hwankoo Kim∗

Division of Computer and Information Engineering, Hoseo University, Asan 31499, Republic of Korea

e-mail : hkkim@hoseo.edu

Abstract. 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] de-fined by A{∗}=T

P ∈∗w-Max(D)ADP[X] and A [∗]

= (T

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.

1. Introduction

Let D be an integral domain with quotient field K, and assume that D is not a field, i.e., D 6= 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) =P{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}

* Corresponding Author.

Received February 22, 2020; revised May 18, 2020; accepted May 30, 2020. 2010 Mathematics Subject Classification: 13A15, 13G05, 13F20.

Key words and phrases: star operation, extension of a star operation to the polynomial ring, t-class group, integrally closed, ∗-Noetherian domain.

[∗] and {∗} of ∗ to D[X]; the star operation [∗] (resp., {∗}) on D[X], first studied in [4] (resp., [5]), is defined by A[∗] _{= (}T

P ∈∗w-Max(D)AD[X]P [X]) ∩ AK[X] (resp., A{∗} =T

P ∈∗w-Max(D)ADP[X]) for all A ∈ F(D[X]); hence A

[w]_{= A}w_{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 7→ I∗ of F(D) into F(D) that satisfies the following three conditions for all 0 6= 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 ₌S{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) 6= ∅, D = TP ∈∗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) ⊆ 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≤ ∗2are 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 opera-tions.

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

Throughout D is an integral domain with quotient field K and D 6= 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∗wand D[X]N∗= {

g

f | 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 N a(D, ∗). It is known that I∗w _{= ID[X]}

N∗∩ K = T

P ∈∗fMax(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 ∆ = {Q ∈ Spec(D[X]) | Q ∩ D = (0) or Q = (Q ∩ D)[X] and (Q ∩ D)∗f

( D} . Set S = D[X][Y ] \ (S{Q[Y ] | Q ∈ ∆}) 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 7→ A[∗]_{, is a star} opera-tion 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[∗]= (T

P ∈∗f-Max(D)AD[X]P [X]) ∩ AK[X], (2) A[∗]_{= AD[X]}

N∗∩ AK[X] and (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 ₌T

Q∈[∗]-Max(D[X])AD[X]Q. Also note that AK[X] = T{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[∗] =

(T

P ∈∗f-Max(D)AD[X]P [X]) ∩ (T{Q ∈ Spec(D[X]) | Q ∩ D = (0) and cD(Q)

∗f ₌ D} = (T

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∗=

T

P ∈∗f-Max(D)(AD[X]N∗)P [X]N∗ = T

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). 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 7→ 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{∗}=T

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 g_f = h

u ∈ D[X][Y ]N (∗)∩ K(X). Then ug = f h and gcD[X](u) = f cD[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))[∗]⊆ f D[X], and thus g_f ∈ 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 = g_f ∈ 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 J D[X][Y ]N (∗) = D[X][Y ]N (∗). So u ∈ uD[X][Y ]N (∗)∩K(X) = uJ D[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)

(⊆) u ∈ B ⇒ uJ ⊆ A for some J ∈ f (D) with J∗ = D ⇒ u ∈ uDP[X] = uJ DP[X] ⊆ ADP[X] for all P ∈ ∗f-Max(D) ⇒ u ∈TP ∈∗f-Max(D)ADP[X].

(⊇) For v ∈T

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. 2

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{∗} = T

P ∈∗f-Max(D)ADP[X] ⊆ ( T

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{∗} =T

P ∈∗f-Max(D)QDP[X] ⊆ QDQ∩D[X] ∩ D[X] = Q, and hence Q{∗} = Q. But Q[∗] _{= (}T

P ∈∗f-Max(D)QD[X]P [X]) ∩ QK[X] = (T

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). 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). 2

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 7→ E? of ¯F (D) into ¯F (D) that satisfies the following three conditions for all 0 6= 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 7→ 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 7→ 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) LetS 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}?0 _{= {u ∈ K(X) | uB ⊆ A for some B ∈ S}} for all A ∈ ¯F (D[X]). Picozza proved that ?0 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 ?0-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?0 ⇒ 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?0_. Hence A{∗2} _{= A}?0_{. Thus ?}0_|

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≤ ∗2are 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 = T

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 f A[∗] ⊆ A[∗]_{. Hence by} Corollary 2.2(3), f AD[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]\P DP[X]∩K[X] = DP[X] = DP[X]; so f ∈ DP[X]. Hence each coefficient of f is contained in (T P ∈∗f-Max(D)DP[X]) ∩ K = T 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(J D[X]){∗} = uJ∗w_{D[X] ⊆} J∗w_{D[X] = (J D[X])}{∗}_{. Since J D[X] is finitely generated, we have u ∈ (D[X])}[{∗}]_. Hence D[∗w]_{⊆ (D[X])}[{∗}]_{, and thus D}[∗w]_{[X] ⊆ (D[X])}[{∗}]_. 2

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)∗w_{D[X] = D[X] (cf., Corollary 2.4(3) for the third equality). Thus the map} ϕ : Cl∗(D) → Cl{∗}(D[X]), given by [I] 7→ [ID[X]], is well-defined.

Next, let I ∈ Inv∗(D) such that (ID[X]){∗} is principal. Since (ID[X]){∗} = I∗w_{D[X], we have I}∗w_{D[X] = f D[X] for some f ∈ I}∗w_{D[X]. But, since I}∗w_{D[X] ∩} K = I∗w _{6= (0), we have f ∈ K, and hence I}∗w _{= I}∗w_{D[X] ∩ K = f D[X] ∩ K =} (f D)∗w _{= f D. Thus, ϕ is injective. Therefore, Cl}∗_{(D) ⊆ Cl}{∗}_(D[X]).

(Proof of Cl{∗}(D[X]) ⊆ Cl[∗](D[X])) This follows because {∗} ≤ [∗] by

Corol-lary 2.4. 2

Theorem 3.2. Cl∗_{(D) = Cl}[∗]_{(D[X]) if and only if D is integrally closed.}

Proof. (⇒) For α = a_b ∈ K such that a, b ∈ D and α is integral over D, let f = bX − a and Q = f K[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 6= c ∈ I, c = qu for some q ∈ Q; so u = c

q. Since f ∈ Q, we have fc_q ∈ 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 gc_q ∈ ID[X]. Hence gc

q = Pn i=0ciX i _{∈ ID[X].} Then gc =Pn i=0(ciq)X i_{. Since each qc}

i∈ qID[X] = Qc, there exist some ai, bi∈ D such that ciq = (biX − ai)c and biX − ai ∈ Q. Hence gc =Pn_i=0Xi(biX − ai)c, and so g =Pn

i=0X i_(b

iX − ai). Note that biα − ai = 0 for each i, and so α = a_bi

i. Since g is monic, bn = 1, and thus α =a_bn

n = 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 = f ID[X] for some f ∈ D[X] and a t-invertible t-ideal I of D [8, Lemma 3.3]. Note that A = A[∗] = (f ID[X])[∗] = f I∗w_{D[X] by Theorem 2.1(4).} Hence I is a ∗-invertible ∗-ideal, and [A] = [ID[X]] = ϕ([I]). Thus ϕ is surjective.2 Corollary 3.3.([8, Theorem 3.6]) Clt_{(D) ∼= Cl}t_{(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]) =

An integral domain D with quotient field K is said to be seminormal if α ∈ K with α2_{, α}3 _{∈ D implies that α ∈ D. Then the following theorem is well known:} P ic(D) ∼= P ic(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 6∈ D, but α2_{, α}3_{∈ D. Consider the fractional ideals} I := (α2_{, 1 + αX) and J := (α}2_{, 1 − αX) of D[X]. Then IJ = (α}4_{, α}2_{+ α}3_{X, α}2₋ 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. 2

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

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

References

[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 exten-sions 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. Alge-bra, 15(1987), 2349–2370.

[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, Pr¨ufer 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.