ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

We introduce the notion of ultra star operation on ultraproduct of integral domains as a map from the set of induced ideals into the set of induced ideals satisfying the traditional properties of star operations. A case of special interest is the construction of an ultra star operation on the ultraproduct of integral domains Ri’s from some given star operations ?i on Ri’s. We provide the ultra b-operation and the ultra v-operation. Given an arbitrary star operation ? on the ultraproduct of some integral domains, we pose the problem of whether the restriction of ? to the set of induced ideals is necessarily an ultra star operation. We show that the ultraproduct of integral domains Ri’s is a ?-Prüfer domain if and only if Ri is a ?i-Prüfer domain for U-many i. Mathematics Subject Classification (2010): 13A15, 13A18, 16W50


Introduction
Let R be an integral domain with quotient field K. Throughout this paper F (R) denotes the set of all nonzero fractional ideals of R and f (R) denotes the set of all nonzero finitely generated fractional ideals of R.
A star operation on R is a mapping A → A of F (R) into F (R) such that for all A, B ∈ F (R) and for all a ∈ K \ {0}, For an overview of star operations, the reader may refer to [5,Sections 32 and 34]. An I ∈ F (R) is a -ideal if I = I. An obvious example of a star operation is the identity map traditionally denoted by d and defined as A d = A for all A ∈ F (R). Another well known star operation we intensively use in this paper is the v-operation. For A, B ∈ F (R), (B : A) = {x ∈ K : xA ⊆ B} and A −1 = (R : A).
The v-operation is defined by A v = A −1−1 for all A ∈ F (R). The ideal A v is also called the v-closure of A. If A v = A, then A is called a divisorial ideal of R. Another

ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS
207 key example of star operation to be used in the paper is the so called b-operation defined on an integrally closed domain R by A b = α AV α for all A ∈ F (R), where {V α } is the family of all valuation overrings of R.
The notion of star operation in commutative rings is often used to not only provide generalizations of classical domains, but to also produce a common treatment and deeper understanding of those domains. One of these instances is the notion of Prüfer -multiplication domain which generalizes the notion of Prüfer domains [3,6]; a notion that will be used in this paper. Under the same spirit, in this paper, we introduce the notion of ultra star operation on an ultraproduct of integral domains and provide some examples. We show that a "true" star operation on a domain does not always restrict to an ultra star operation. For easy reference, in Section 2, we review some definitions and results of ultraproducts of commutative rings that can be found in [10,11] and provide some results to be used in later sections.
In Section 3, we define an ultra star operation on the ultraproduct R of integral domains R i 's and show how to build a "star operation", called ultra star operation, from star operations on R i 's. We show that this construction yields the ultra voperation and the ultra b-operation when we start with the traditional v-operations on the R i 's and the traditional b-operations on the R i 's respectively.
In Section 4, we show as an application of the introduction of the notion of ultra star operation that if an ultra star operation on R is built from star operations i 's on the components R i 's as in Proposition 3.1, then R is a -Prüfer domain if and only if R i is a i -Prüfer domain for U-many i. We recall here that if U is an ultrafilter on a nonempty set X, a property P holds for U-many i if the set of all i ∈ X such that R i satisfy P is an element of U (see Section 2 below).

Notations and preliminaries
Let X be a set. Let P(X) be the collection of all subsets of X. A nonempty set F ⊆ P(X) is a filter on P(X) if for all A, B ∈ P(X) we have: F is a maximal filter if the only filter that contains it is P(X). By Zorn's Lemma, every proper filter of P(X) is contained in a maximal filter of P(X).
An ultrafilter U on X is a maximal filter on P(X). Equivalently, a filter U on X is an ultrafilter if and only if for all A ⊆ X, either A ∈ U or X \ A ∈ U.
If an ultrafilter U contains a finite subset of X, then it contains a singleton set, 208 OLIVIER A. HEUBO-KWEGNA say {x}, and x is an element of every element of U. In this case U is a principal ultrafilter. An ultrafilter that is not principal is called a free ultrafilter. Let {R i } i∈I be the collection of commutative rings indexed by a set I. The direct-product R of the commutative rings {R i } i∈I is a commutative ring consisting of elements If U is an ultrafilter on I, then we define on R a relation ∼ by The quotient set R/ ∼ is denoted U R i =: R is called the ultraproduct of the R i 's with respect to the ultrafilter U. We denote simply by (a i ) the element of R determined by the equivalence class of (a i ). In the case that for all i ∈ I, R i = R, then R is called the ultrapower of R and is denoted R U .

Los' Theorem.
A property P holds for U-many i if the set of all i such that R i satisfy P is an element of U. Any first order formula over the language of commutative rings utilizes the symbols +, ., =, 0, 1. One of the most fundamental properties of ultraproducts due to Los is that they preserve first order properties. It follows from Los' Theorem that R is an integral domain (a field resp.) if and only if R i is an integral domain (a field resp.) for U-many i. This is because the theory of integral domains and fields consists of finitely many axioms, all of which can be easily be expressed in the first order language of integral domains and fields.
If U is a principal ultrafilter, say {i} ∈ U, then it is not hard to see that R ∼ = R i . Thus, throughout this paper, we are interested in a collection of integral domains {R i } i∈I indexed by an infinite set I, and U is a non principal ultrafilter. The quotient field of each R i is denoted F i . The quotient field of R is the ultraproduct An ideal A of a commutative ring R is definable in R if there exists a first order formula ψ(x, y 1 , . . . , y n ) in the language of commutative rings and r 1 , . . . , r n ∈ R such that r ∈ A ⇔ ψ(r, r 1 , . . . , r n ) is true in R.
A finitely generated ideal A of a domain R with quotient field K and its dual A −1 are definable, as well as the maximal ideal of any quasilocal ring (see [10]). For example, if A := (r 1 , r 2 , . . . , r n ) is a finitely generated ideal of R, then x ∈ K is in

Recall that an ideal
Some facts about the algebra of induced ideals are provided in Proposition 2.2. Recall also that an ideal A of a commutative ring is n-generated if A can be generated by n elements. Note that an ideal A is n-generated if and Proof. All the statements are shown in [10, Lemma 2.1] except (vi). To see (vi), In general, except in some very special cases, an ultraproduct of Noetherian rings is non-Noetherian (see for example [ We next assume that the R i 's are Noetherian and are chosen such that the ultraproduct R is Noetherian. Then we aim to prove that an induced ideal J of R is integrally closed if and only if each component J i of J is an integrally closed ideal of R i for U-many i. Recall that for an ideal A of an integral domain R, the integral closure A b of A consists of r ∈ R satisfying the integral dependence equation of r over A: r s + a 1 r s−1 + a 2 r s−2 + . . . + ra s−1 + a s = 0 for some positive integer s and

Observe that an induced ideal A = (A i ) U is integrally closed if and only if
A satisfies the sentence φ n that asserts that if a, b ∈ A and f (a, b) = 0 for some degree n and for some f (x, y) := x n +r n−1 x n−1 y+. . .+r 0 y n ∈ A[x, y].

Ultra star operations on ultra product of domains
In the star operation context, it is customary to replace the v-operation by an arbitrary star operation in results such as Proposition 2.2 (viii) in order to generalize concepts. Proposition 2.2 (viii) is therefore our motivation to introduce the notion of induced star operation on an ultraproduct of integral domains.
From now on assume that i is a star operation on R i for each i ∈ I and denote Ind(R) the collection of induced ideals of R. Defined the map: : For (Ind 1 ), note that R = (R i ) U and hence R = R, since R i i = R i . Note also that ((a i )A) U = (a i A i ) U and it will then follow that ((a i )A) = (a i )A , since We have then motivated the following definition:

2.
The v-operation on R preserves the induced ideals. In fact, by Proposition

(vi), we have
So the v-operation is a strong ultra star operation on R.

In Proposition 3.1, if each
is an ultra star operation that coincides with the restriction of the v-operation of R on the set of induced ideals in view of Proposition

(vi) and (viii). So the mapping
where each R i is assumed to be integrally closed so that each b i is a star operation on R i , then the mapping A = (A i ) U → A = (A bi i ) U is an ultra star operation. Note that by Proposition 2.3, A is an integrally closed ideal of R (we do not need the assumption of the R i 's being Noetherian for this) and therefore can be legitimately labeled as the ultra b-operation.
Also note that R is integrally closed as all the R i 's are integrally closed. In fact, recall that a domain R is integrally closed if and only if for all n > 0, R satisfies the sentence φ n that asserts that if a, b ∈ R and f (a, b) = 0 for some degree n form f (x, y) := x n + r n−1 x n−1 y + . . . + r 0 y n ∈ R[x, y], then b divides a ∈ R. Thus the statement holds by Los' Theorem.
Note however that we do not know whether the ultra b-operation coincide with the restriction of the b-operation of R. More generally, we do not expected all star operations on R to preserve induced ideals and therefore we may not obtain an ultra star operation just by restricting any star operation of R on the set of induced ideals. We can make the following observation for the b-operation in the case we assume that the R i 's are Noetherian domains such that R is Noetherian:

the later equality holds by combining Proposition 2.2 (i) and Proposition 2.3. If
Question: If A is an induced ideal of R, must A b be an induce ideal?

Ultra star operations and Prüfer -multiplication domains
In this section we aim to establish that if an ultra star operation on R is built from star operations on the components R i 's as in     Proof. Suppose that A is f -invertible. From the above observation, it follows that A is -invertible, i.e., (AA −1 ) = R. We have, for each maximal f -ideal M , AR M is invertible and therefore principal.    prove that (iii) ⇒ (i). Let x, y ∈ R, note that if P is a prime ideal of R, we have xR P + yR P = (a, b)R P for some a, b ∈ R. But if P is a f -maximal ideal of R then, by Lemma 4.2 and by hypothesis, (a, b)R P is principal, that is, R P is a valuation domain.
Remark 4.4. Note that the above proof of the preceding lemma is a general version of the proof for the PvMD case (see [8,Lemma 1.7]). Now that we have a bound on finitely generated ideals to characterize P MD, we return to ultraproduct. Recall that an ideal J of a commutative ring is n-generated if J can be generated by n-elements. Then an n-generated ideal A is a -invertible ideal of R if and only if A i is an n-generated i -invertible ideal of R i for U-many i.
Proof. Suppose that A = (A i ) U is an n-generated -invertible ideal of R, that is, (AA −1 ) = R. Note that since A is n-generated, A i is n-generated for U-many i. So

Conversely suppose that
Remark 4.6. Let i 's be star operations on R i 's. Consider ( i ) f be the star operation of finite type associated with each i . If we denote f the ultra star operation on R built from the ( i ) f 's (note that f is just a notation, so does not mean f is the finite type associated to ), then using Lemma 4.5, we have that an n-generated ideal A is a f -invertible ideal of R if and only if A i is an n-generated ( i ) f -invertible ideal of R i for U-many i.