Polynomial closure in essential domains

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1 manuscripta math. 117, (2005) Springer-Verlag 2005 Mi Hee Park Francesca Tartarone Polynomial closure in essential domains Received: 3 April 2004/Revised version: 5 February 2005 Published online: 15 April 2005 Abstract. Let D be a domain, E a subset of its uotient field K, and Int(E, D) ={f K[X] f(e) D}. The polynomial closure of E is the set cl D (E) ={x K f(x) D, f Int(E, D)}. We compare the polynomial closure with the divisorial closure in a general setting and then in an essential domain. Especially, we show that these two closures of ideals are the same if D is a Krull-type domain. Introduction Let D be an integral domain with uotient field K. For each subset E K, Int(E, D) := {f K[X] f(e) D} is called the ring of D-integer-valued polynomials over E. As usual, when E = D, we set Int(D) := Int(D, D). The polynomial closure (in D)ofE is defined as the set cl D (E) := {x K f(x) D, f Int(E, D)}, that is, cl D (E) is the largest subset F K such that Int(E, D) = Int(F, D). The first papers about polynomial closure (for instance, [3], [8], [10] and [12]) set it in a topological context. Among many other results, it was proven that in a Dedekind domain with finite residue fields, the polynomial closure of a subset E is the same as the intersection of its topological closures in every maximal ideal-adic topology. There is another way to look at the polynomial closure. That is to study it as a star-operation [7, Lemma 1.2]. Recall that a fractional ideal I of D is a D-module such that di D for some nonzero element d D, and denote by F(D) the set of nonzero fractional ideals of D. A star-operation is a mapping : F(D) F(D), I I, satisfying the following properties for each a K\{0} and I,J F(D): ( 1) (ad) = ad; (ai) = ai ; ( 2) I I ; This work was done while the first author was visiting KIAS. She would like to thank KIAS for the hospitality and support. M.H. Park: Department of Mathematics Chung-Ang University Seoul , Korea. mhpark@cau.ac.kr F. Tartarone: Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, Roma, Italy. tfrance@mat.uniroma3.it DOI: /s

2 30 M.H. Park, F. Tartarone ( 3) I J I J ; ( 4) (I ) = I. The best known star-operation is the v-closure (or, the divisorial closure): I I v := (I 1 ) 1, where I 1 :={x K xi D}. A nonzero fractional ideal I is called divisorial if I = I v. The divisorial closure has a prominent position in this context; it is the maximal star-operation in the sense that I I v for each I F(D) and each star-operation. In Section 1, we point out some interesting analogies between the polynomial closure and the divisorial closure. We observe that the divisorial closure of a fractional ideal I can be defined in the same way as the polynomial closure is defined just replacing the set of polynomials Int(I, D) by I 1 X. Motivated by this observation, we introduce a new star-operation, Cl D ( ), which is constructed like the polynomial closure but using the set of polynomials D[X/I] := a I\{0} D[ X a ] instead of Int(I, D). When D[X/I] = Int(I, D) (which happens, for example, whenever Int(D) = D[X]), obviously Cl D (I) = cl D (I). In general, I 1 X D[X/I] Int(I, D), whence cl D (I) Cl D (I) I v. In Section 2, we prove that Cl D ( ) is eual to the divisorial closure in any essential domains (Proposition 2.4). Moreover, Cl D ( ) has turned out to be an useful tool to study the euivalence between the polynomial closure and the divisorial closure in this class of domains. The study about the polynomial closure in this direction was developed by M. Fontana et al. in [7]. They proved, for instance, that in a valuation domain the polynomial closure and the divisorial closure of a fractional ideal are the same. We prove, more generally, that these two closures coincide in a large class of essential domains, the Krull-type domains, which include Krull domains and semi-uasi-local Prüfer domains (Theorem 2.8). 1. A new star-operation Cl D ( ) Let I be a nonzero fractional ideal of D and define I 1 ={x K xi D}. Every D-homomorphism from I to D can be uniuely extended to a D-homomorphism from K to K. Henceforth, we identify Hom D (I, D) with the subset of Hom D (K, K) mapping I into D. For each f Hom D (K, K),ifweleta = f(1), then f is the multiplication on K by a. Thus we have a canonical isomorphism of D-modules ϕ : I 1 Hom D (I, D) defined by ϕ(a)(x) = ax for all a I 1 and x I. Note that each function ϕ(a) is a polynomial function induced by the polynomial ax. By identifying I 1 with Hom D (I, D), we also have a canonical isomorphism λ : I v Hom D (Hom D (I, D), D) defined by λ(x)(f ) = f(x)for all x I v and f Hom D (I, D) [11, page 37].

3 Polynomial closure in essential domains 31 Thus we have I v ={x K f(x) D, f Hom D (I, D)} ={x K f(x) D, f I 1 X}. Recalling that cl D (I) ={x K f(x) D, f Int(I, D)}, we can see that the divisorial closure I v is a sort of polynomial closure obtained by substituting I 1 X for Int(I, D). From the inclusion I 1 X Int(I, D), it directly follows the fact that I v cl D (I) [7, Corollary 1.3 (4)]. Another interesting analogy between the two types of closure is that as cl D (I) is the largest fractional ideal of D such that Int(I, D) = Int(cl D (I), D), soi v is the largest fractional ideal of D such that Hom D (I, D) = Hom D (I v,d). Borrowing the above idea, that is, using a suitable subset of polynomials, we will construct a new star operation later in this section. To begin with, we recall the following result: Lemma 1.1. [6, Lemma 4.5] Let D be a domain such that Int(D) = D[X]. Then, for each nonzero D-submodule E of K, we have Int(E, D) = D[X/E] := a E\{0} D[ X a ]. Moreover, D[X/E] is a graded ring of the following type D[X/E] = D ( u E\{0} 1 D)X ( u u E\{0} 1 u n D)Xn. Let I be a nonzero fractional ideal of a domain D. We let I (n) denote the D-module generated by the set {u n u I}. Proposition 1.2. Let D be a domain such that Int(D) = D[X]. ForI,J F(D), cl D (I) = cl D (J ) if and only if I (n) v = J (n) v for each n 0. Proof. From Lemma 1.1, Int(I, D) is a graded ring of the form Note that ( n 0 u I\{0} u I\{0} 1 u n D)Xn. 1 u n D = I (n) 1. The same holds for Int(J, D). Thus, Int(I, D) = Int(J, D) if and only if I (n) 1 = J (n) 1, that is, I (n) v = J (n) v for each n 0. Remark 1.3. Let D be a domain such that Int(D) = D[X]. If I is a nonzero fractional ideal of D, then by Lemma 1.1 and [1, Corollary 1, II.1.6], we have the canonical isomorphisms

4 32 M.H. Park, F. Tartarone Hom D (Int(I, D), D) = Hom D (D[X/I],D) = Hom D (D I 1 X I (n) 1 X n,d) = Hom D (I (n) 1 X n,d) n 0 = n 0 Hom D (I (n) 1,D) = n 0 I (n) v. Let θ : Hom D (Int(I, D), D) n 0 I (n) v be the isomorphism defined by the above series of natural isomorphisms and let ψ:cl D (I) Hom D (Int(I, D), D) be the injective map defined by ψ(x)(f ) = f(x) for all f Int(I, D). Then the composite map θψ :cl D (I) n 0 I (n) v is injective and it is given by x (1,x,x 2,x 3,...). (For the better understanding, we describe the isomorphism θ 1 specifically: Let x = (x 0,x 1,x 2,...) n 0 I (n) v. Then θ 1 (x) is given by f = n i=0 a i X i n i=0 a i x i.) Proposition 1.4. Let D be a domain such that Int(D) = D[X]. IfI is a nonzero fractional ideal of D, then cl D (I) ={x K x n I (n) v, n 0}. Proof. It follows from the map θψ mentioned above. Here is a more elementary proof. Let x cl D (I). Since I (n) 1 X n Int(I, D) = D[X/I], by the definition of polynomial closure, I (n) 1 x n D. Thus x n I (n) v for each n 0. Conversely, let x K such that x n I (n) v for each n 0. Then I (n) 1 x n D. So, for an arbitrary polynomial f(x)= a 0 + a 1 X + +a n X n Int(I, D) = D[X/I] = n 0 I (n) 1 X n,wehavef(x)= a 0 + a 1 x + +a n x n D. Thus x cl D (I). Motivated by Proposition 1.4, we define a new star-operation without the assumption that Int(D) = D[X]. Definition 1.5. For a nonzero fractional ideal I of D, weset Cl D (I) := {x K x n I (n) v, n 0} ={x K f(x) D, f D[X/I]}. Lemma 1.6. With the above notation, we have (1) the map Cl D ( ) : F(D) F(D), I Cl D (I), is a star-operation; (2) for each I F(D), cl D (I) Cl D (I) (the euality holds when Int(D) = D[X]). Proof. (1) It is an easy exercise to check that all the properties reuired for a star-operation are satisfied. (2) Since D[X/I] Int(I, D), it is obvious that cl D (I) Cl D (I). In particular, if Int(D) = D[X], then Int(I, D) = D[X/I] by Lemma 1.1 and hence cl D (I) = Cl D (I).

5 Polynomial closure in essential domains Polynomial closure and divisorial closure in essential domains Let D be an integral domain with D = D P for some subset P Spec(D). The P-polynomial closure of I F(D) is P-cl D (I) := cl DP (ID P ). The mapping I P-cl D (I) defines a star-operation on D with P-cl D (I) cl D (I) [7, Lemma 1.4]. In [7] M. Fontana et al. study the polynomial closure in essential domains and compare it with other star-operations including the one just mentioned. In this section, we carry on this studying. We recall some relevant definitions. An essential domain is a domain D such that D = D P, (2.1) where P Spec(D) and each D P is a valuation domain. A domain D is Krull-type if it is essential and the intersection (2.1) is locally finite (that is, each nonzero element x K is a unit in D P for all but finitely many P P), and a Krull-type domain is strong Krull-type if the valuation domains D P are pairwise independent (that is, each pair of valuation domains D P doesn t have common overrings except K) [9, 43]. By [7, Theorem 1.9], if D is an essential domain with the representation (2.1), then we have I t P-cl D (I) cl D (I) I v for each nonzero fractional ideal I. IfD is strong Krull-type, then P-cl D (I) = cl D (I) for each I F(D) and under some conditions on the set of t-maximal ideals of D, cl D (I) = I v [7, Corollary 1.12]. We strengthen these results in the following. Remark 2.1. We will freely use the following facts: (a) Let D be a domain, I an ideal of D and A an overring of D. Then, (A: I) = (A: IA). (b) If D = λ D λ, where D λ are overrings of D, and I is a nonzero fractional ideal of D, then (D : I) = (D λ : I) = λ : ID λ ). λ λ (D (c) Let V be a valuation domain. Then each prime ideal P V is divided, that is, P = PV Q for each prime ideal Q of V such that P Q [9, Theorem 17.6 (b)]; a nonzero ideal I of V is invertible if and only if it is principal [9, Theorem 17.1 (1)]; moreover, if I is a prime ideal, then it is maximal [9, Theorem 17.3 (a)].

6 34 M.H. Park, F. Tartarone Theorem 2.2. Let D = D P, P Spec(D), be a Krull-type domain. Then, P-cl D (I) = I v for all I F(D) if and only if D is a strong Krull-type domain. Proof. Let I be a nonzero integral ideal of the Krull-type domain D. We claim first that (D : I)D Q = ( ) (DP : I)D Q (2.2) for all Q P. The containment (D : I)D Q ( ) (DP : I)D Q is clear. For the reverse inclusion, take ξ ( ) (DP : I)D Q \{0}. Then ξ is a unit in DP for all but finitely many P P, say P 1,...,P r. Since ξ (D Pi : I)D Q, there exists s i D \ Q such that s i ξ (D Pi : I). Let s = s 1 s r. Then s D \ Q and sξi D Pi for all i = 1,...,r. It follows that sξi D P = D, whence sξ (D : I) and ξ (D : I)D Q. Now assume that D is a strong Krull-type domain. Since the D P s are pairwise independent valuation domains, D P D Q = K for any two distinct prime ideals P and Q in P. Thus we have (D : I)D Q = ( ) ( ) (DP : I)D Q = (DP : I)D P D Q = (D Q : I) = (D Q : ID Q ). Therefore, I v = (D : (D : I)) = (D P : (D : I)) = (D P : (D : I)D P ) = (D P : (D P : ID P )) = (ID P ) v = cl DP (ID P ) = P-cl D (I), where the last second euality follows from [7, Proposition 1.8]. For the converse, we assume that P-cl D (I) = I v for each I F(D) but D is not a strong Krull-type domain. For each distinct prime ideals P, Q in P, let (P,Q) be the prime ideal of D such that D P D Q = D (P,Q). In other words, (P,Q) is the prime ideal maximal among all prime ideals contained in P Q. Note that (P,Q) is (0) if and only if D P and D Q are independent. Since D is not a strong Krull-type domain, there exists a prime ideal P 0 P such that D P0 and D Q are dependent for some Q( P 0 ) P. Put := Q P\{P 0 } (P 0,Q). Since D P0 is a valuation domain, the set of prime ideals {(P 0,Q) Q P \{P 0 }} is linearly ordered by inclusion and hence is a nonzero prime ideal of D contained in P 0. Moreover, since the intersection D = D P is locally finite, there exist finitely many prime ideals P 1,...,P n P (n 1) such that = (P 0,P 1 ) = = (P 0,P n ) and (P 0,Q) for all Q P \{P 0,...,P n }.

7 Polynomial closure in essential domains 35 Let a be a nonzero element of. Then a is contained in only finitely many prime ideals of P, say P 0,...,P n,p n+1,...,p n+m. Note that P n+i for all i = 1,...,m. Choose b \ m j=1 P n+j and put I := (a, b)d P0 D. Then ID P0 = (a, b)d P0.Fori = 1,...,n,wehave ID Pi = (a, b)d P0 D Pi D Pi = (a, b)d D Pi = (a, b)d, because (a, b)d D = D Pi D Pi.Fori = n+1,...,n+m, ID Pi = D Pi, because b I \ P i.forp P \{P 0,...,P n+m }, ID P = D P, because a I \ P. Note that since D P0 is a valuation domain, ID P0 = (a, b)d P0 is principal, say cd P0 (where c = a or b) and so (a, b)d = (a, b)d P0 D = cd. Then we have ( n ) I 1 = (D P0 : ID P0 ) (D Pi : ID Pi ) (D P : ID P ) i=1 ( n ) = (D P0 : cd P0 ) (D Pi : cd ) = 1 c D P 0 ( n i=1 i=1 ) 1 c D P i ( = 1 n ) D P0 D Pi c i=1 ( 1 D ) D P = 1 c c, \{P 0,...,P n } \{P 0,...,P n } \{P 0,...,P n } \{P 0,...,P n } D P cd P (D P : ID P ) where the last second inclusion follows from the fact D Pi = D D Pi, i = 1,...,n. Hence, c 1 I v = P-cl D (I) cl DP0 (ID P0 ) = (ID P0 ) v = cd P0. Thus we have 1 D P0, that is, (D : )D P0 = D P0. Meanwhile, by (2.2), (D : )D P0 = ( ) (DP : D P )D P0 ( n ( ) ) = (DPi : D Pi )D P0 i=0 ( n ) = D D P0 i=0 = D = D, \{P 0,...,P n } \{P 0,...,P n } D (P,P0 ) \{P 0,...,P n } D P D P0 ( ) (DP : D P )D P0

8 36 M.H. Park, F. Tartarone where the third euality follows from [5, Corollary 3.1.5] and [9, Theorem 7.6], and the last euality follows from the fact that (P,P 0 ) for all P P \{P 0 }. Thus we have D P0 = D, which contradicts the fact that P 0. Corollary 2.3. If D is a strong Krull-type domain, then cl D (I) = I v for all I F(D). Proposition 2.4. Let D = D P be an essential domain. Then Cl D (I) = I v for all I F(D). Proof. Let I be a nonzero integral ideal of D. By Proposition 1.2, Cl D (I) = I v if and only if (I (n)) v = (I v (n)) v for all n 1. Put I := ID P. Then the mapping I I defines a star-operation on D, whence I I I v and (I ) v = I v [9, Theorem 32.5 and Theorem 34.1]. We claim that I (n)d P = I n D P for all n 1 and all P P. Let a 1,...,a n I. Since D P is a valuation domain, we may assume that a 1 D P a 2 D P a n D P. Then a 1 a n a n 1 D P I (n)d P. Therefore I n D P I (n)d P. The reverse inclusion I (n)d P I n D P is obvious. Therefore, I (n) = I (n)d P = I n D P = (I n ), and so we have (I v (n)) v (I (n)) v = ( I (n) ) v = ( (I n ) ) v = (I n ) v = ( (I v ) n) v (I v(n)) v. Thus I (n) v = (I v (n)) v for all n 1 and Cl D (I) = I v. Corollary 2.5. Let D = D P be an essential domain such that Int(D) = D[X]. Then cl D (I) = I v for all I F(D). Proof. This directly follows from Lemma 1.6 and Proposition 2.4. Lemma 2.6. [4, Proposition 1.8 (b) and Corollary 1.9] Let D be a domain with a nonzero divided prime ideal. Then the map I I establishes a one-toone correspondence between the set of all the fractional ideals of D such that I I v D and the set of all the nonzero fractional ideals of D. Furthermore, if I F(D) such that I I v D, then ( ) I 1 = I 1 and ( ) I = I v v. Proposition 2.7. Let D be a semi-uasi-local Prüfer domain. Then cl D (I) = I v for all I F(D).

9 Polynomial closure in essential domains 37 Proof. Let P 1,...,P n be the maximal ideals of D. We use the induction on n. If n = 1, then D is a valuation domain, so the conclusion follows from [7, Proposition 1.8] (or Corollary 2.3). Assume that n>1 and that the theorem holds for Prüfer domains with at most (n 1) maximal ideals. By rearrangements, if necessary, we may assume that there exists a positive integer r( n) such that D Pi and D P1 are dependent for i r, and D Pi and D P1 are independent for i>r. Put S := r i=1 D Pi and T := n i=r+1 D Pi (where T := K,ifr = n). So D = S T. Note that they are Prüfer domains with r and n r maximal ideals, respectively [9, Theorem 22.8]. Case 1. Assume that r<n. By induction hypothesis, the theorem holds for S and T. So, in particular, for each nonzero integral ideal I of D, cl S (IS) = (IS) v and cl T (IT ) = (IT ) v. Since S = D S0 and T = D T0 for the multiplicative subsets S 0 = D \ r i=1 P i and T 0 = D \ n i=r+1 P i [9, Lemma 5.4], we have that cl S (IS) = cl S (I) and cl T (IT ) = cl T (I) [3, Lemma 3.4]. So (IS) v = cl S (I) and (IT ) v = cl T (I). We observe that ST = K. Actually, since (P 1,P i ) (0) for all i r and (P 1,P j ) = (0) for all j r +1, we have (P i,p j ) = (0) for i r and j r +1. (Here, we use the same notation as in the proof of Theorem 2.2.) This implies that there does not exist nonzero prime ideals of D such that ST D, and hence by [9, Theorem 26.1], ST = K. Therefore, and similarly, Thus we have (D : I)S = (D : I)D S0 = ((S : I) (T : I))D S0 = (S : I)D S0 (T : I)D S0 = (S : I) (T : I)TS = (S : I) = (S : IS), (D : I)T = (T : I) = (T : IT). I v = (D : (D : I)) = (S : (D : I)) (T : (D : I)) = (S : (D : I)S) (T : (D : I)T) = (S : (S : IS)) (T : (T : IT)) = (IS) v (IT ) v = cl S (I) cl T (I) = cl DS0 (I) cl DT0 (I) cl D (I), where the last inclusion follows from [2, Lemma IV.2.1]. Hence I v = cl D (I). Case 2. Assume that r = n, so that (P 1,P i ) (0) for each i = 2,...,n. Let := n i=2 (P 1,P i ). Then is a nonzero prime ideal of D and = (P 1,P j ) for some j 2. Without loss of generality, we may assume that = (P 1,P n ). Since is contained in all the maximal ideals of D, is a divided ideal, that is, D = (in fact, D = D Pi for all i = 1,...,n, whence D = n i=1 D Pi = ). So, for each nonzero integral ideal I of D, I and are comparable. In fact, if I, then ID Pi D Pi, for all i = 1,...,n.AsD Pi being a valuation domain, = D Pi ID Pi for all i = 1,...,n, whence n i=1 ID Pi =I. Case 2.1. Assume that I. Then by Lemma 2.6, ( I ) v = I v. Note that D is a semi-uasi-local Prüfer domain with n-maximal ideals P 1,, P n. Moreover,

10 38 M.H. Park, F. Tartarone ( D ) D P1 P 1 = and ( D ) = D Pn Pn are independent. So, by Case 1, ( I ) v = cl D ( I ). Since is nonmaximal, has infinite residue field, whence Int(D) D [X] [2, Proposition I.3.4]. Therefore, by [7, Proposition 3.2], cl D ( I ) = cl D(I). Thus we have I v = ( I ) v = cl D ( I ) = cl D(I), and hence I v = cl D (I). Case 2.2. Assume that I and ID is not a principal ideal of D, hence it is not invertible. Then II 1 ID (ID ) 1 D =. So, (ID ) 1 ( : ID ) ( : I) (D : I) = I 1 (ID ) 1. Thus we have (ID ) 1 = ( : I) = I 1. Since I n D is not invertible for all n 1, similarly we have (I n D ) 1 = (D : I n D ) = (D : I n ) = (I n ) 1. Then, by the same argument used in the proof of Proposition 2.4, I (n)d = I n D and so I (n) 1 = (D : I (n)) (D : I (n)d ) = (D : ID (n)) = (D : I n D ) = (D : I n ) = (I n ) 1 I (n) 1. Now we claim that Int(I, D) = D[X/I]. By [6, Lemma 4.5] and [3, Lemma 3.4], D[X/I] Int(I, D) Int(I, D ) = Int(ID,D ). Since Int(D ) = D [X], again by [6, Lemma 4.5], Int(ID,D ) = D [X/ID ]. From the above euations, it follows that Thus we have D [X/ID ] = n 0 (ID (n)) 1 X n = D I (n) 1 X n n 1 = D + D[X/I]. D[X/I] Int(I, D) D + D[X/I]. But, Int(I, D) K = D, so that Int(I, D) = D[X/I]. Therefore, cl D (I) = Cl D (I), and hence cl D (I) = I v by Proposition 2.4. Case 2.3. Assume that I, ID = ad for some a I, buti v ad. Put J := a 1 I. Then since D J J v D, J is a nonzero fractional ideal of D. By the same argument as in Case 2.1, we have J v = cl D (J ), from which it follows that I v = cl D (I).

11 Polynomial closure in essential domains 39 Case 2.4. Assume that I, ID = ad for some a I, and I v = ad. Put J := a 1 J I. Then J v = D, and hence by Lemma 2.6, and J v are not fractional ideals of D. Since D isaprüfer domain, it is integrally closed and so Int( J, D ) = D = Int( J v, D ) by [2, Proposition I.1.9]. Moreover, [X] D[X/D ] Int(D,D)= Int(J v,d) Int(J, D) Int(J, D ) = Int(J D,D ) = Int(D ) = D [X]. Considering the canonical map we have : D [X] ( D D )[X] = ( D )[X], Int(J v,d) [X] = (Int(J v,d)) = Int( J v, D ) = D and Int(J, D) = (Int(J, D)) [X] = Int( J, D ) = D by [2, Proposition I.3.8 and Remark I.3.9] and [7, Lemma 3.1].Therefore, Int(J v,d) = D + [X] = Int(J, D), which implies that J v = cl D (J ). Therefore, I v = cl D (I). Theorem 2.8. Let D = D P be a Krull-type domain. Then cl D (I) = I v for all I F(D). Proof. Let I be a nonzero proper integral ideal of D. Then I is contained in only finitely many prime ideals P 1,...,P n in P. Put S := n i=1 D Pi and T := \{P 1,...,P n } D P. (We may assume that P \{P 1,...,P n }, otherwise D = S is a semi-uasi-local Prüfer domain, so the conclusion follows from Proposition 2.7.) So D = S T. Notice that (T : I) = T. In fact, (T : I) = (D P : I) = (D P : ID P ) = \{P 1,...,P n } \{P 1,...,P n } D P = T. \{P 1,...,P n } We claim that (D : I)S = (S : I). Put S 0 := D \ n i=1 P i. Then S = D S0, and (D : I)S = ((S : I) (T : I)) S = ((S : I) T ) D S0 = (S : I)D S0 T S0 = (S : I) T S0. Since (S : I) (ST : I) = (T S0 : I), to get the euality (D : I)S = (S : I) it suffices to show that (T S0 : I) = T S0. By the local finiteness of the intersection T = \{P 1,...,P n } D P, it follows that T S0 = \{P 1,...,P n } (D P ) S0 [9, Proposition

12 40 M.H. Park, F. Tartarone 43.5]. Moreover, each D P is a valuation domain, so (D P ) S0 = D (P) for some prime ideal (P)contained in P. Note that I (P)for all P P \{P 1,...,P n }. Therefore, (T S0 : I) = (D (P) : I) = D (P) = T S0. Thus we have \{P 1,...,P n } \{P 1,...,P n } I v I v S = (D : (D : I))S (S : (D : I)S) = (S : (S : I)) = (IS) v = cl S (IS) = cl S (I), where the last second euality follows from Proposition 2.7. Also, observe that Int(I, T ) = Int(I, D P ) = Int(ID P,D P ) = \{P 1,...,P n } \{P 1,...,P n } Int(D P ) = Int(T ), \{P 1,...,P n } where the last euality follows from [2, Proposition I.2.5]. In particular, this observation implies that Int(I, T ) = Int(I v,t), because Int(T ) Int(I v,t) Int(I, T ). Finally, we get Int(I, D) = Int(I, S) Int(I, T ) = Int(cl S (I), S) Int(I v,t) Int(I v,s) Int(I v,t)= Int(I v,d) Int(I, D). Thus Int(I, D) = Int(I v,d), and hence cl D (I) = I v. References [1] Bourbaki, N.: Algebra I. Chapters 1 3. Translated from the French. Reprint of the 1989 English translation, Springer-Verlag, Berlin, 1998 [2] Cahen, P.-J., Chabert, J.-L.: Integer-Valued Polynomials. Amer. Math. Soc. Surveys and Monographs, Vol. 48, Providence, 1997 [3] Cahen, P.-J.: Polynomial closure. J. Number Theory 61(2), (1996) [4] Fontana, M., Gabelli, S.: On the class group and the local class group of a pullback. J. Algebra 181, (1996) [5] Fontana, M., Huckaba, J., Papick, I.: Prüfer Domains. Dekker, New York, 1997 [6] Fontana, M., Izelgue, L., Kabbaj, S. Tartarone, F.: On the Krull dimension of domains of integer-valued polynomials. Expositiones Math. 15, (1997) [7] Fontana, M., Izelgue, L., Kabbaj, S., Tartarone, F.: Polynomial closure in essential domains and pullbacks. Advances in Commutative Ring theory, Lecture Notes Pure Appl. Math. Marcel Dekker 205, (2000) [8] Frisch, S.: Substitution and closure of sets under integer-valued polynomials. J. Number Theory 33, (1996) [9] Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York, 1972; rep. Queen s Papers in Pure and Applied Mathematics, Vol. 90, Queen s University, Kingston, 1992

13 Polynomial closure in essential domains 41 [10] Gilmer, R.: Sets that determine integer-valued polynomials. J. Number Theory 33, (1989) [11] Matlis, E.: Torsion-free modules. The University of Chicago Press, Chicago, 1972 [12] McQuillan, D.L.: On a theorem of R. Gilmer. J. Number Theory 39, (1991)