ON THE SPECTRUM OF RINGS OF FUNCTIONS

SOPHIE FRISCH

Abstract. For D a domain and E ⊆ D, we investigate the prime spectrum of rings of functions from E to D, that is, of rings contained in ∏_{e∈E} D and containing D. Among other things, we characterize, when M is a maximal ideal of finite index in D, those prime ideals lying above M which contain the kernel of the canonical map to ∏_{e∈E}(D/M) as being precisely the prime ideals corresponding to ultrafilters on E. We give a sufficient condition for when all primes above M are of this form and thus establish a correspondence to the prime spectra of ultraproducts of residue class rings of D. As a corollary, we obtain a description using ultrafilters, differing from Chabert’s original one which uses elements of the M-adic completion, of the prime ideals in the ring of integer-valued polynomials Int(D) lying above a maximal ideal of finite index.

1. Introduction

Let D be an integral domain, E ⊆ D, and R a subring of ∏_{e∈E} D, containing D. The elements of R can be interpreted as functions from E to D and, consequently, we call R a ring of functions from E to D. We will investigate the prime spectra of such rings of functions. We obtain, for quite general R, a partial description of the prime spectrum, cf. Theorems 3.7 and 5.3, and in special cases a complete characterization, cf. Corollary 6.5.

Our motivation is the spectrum of a ring of integer-valued polynomials: For D an integral domain with quotient field K, let Int(D) = {f ∈ K[x] | f(D) ⊆ D} be the ring of integer-valued polynomials on D. More generally, when K is understood, we let Int(A, B) = {f ∈ K[x] | f(A) ⊆ B} for A, B ⊆ K.

If D is a Noetherian one-dimensional domain, a celebrated theorem of Chabert [1, Ch. V] states that every prime ideal of Int(D) lying over a maximal ideal M of finite index in D is maximal and of the form

M_{α} = {f ∈ Int(D) | f(α) ∈ M̂},

where α is an element of the M-adic completion D̂_{M} of D and M̂ the maximal ideal of D̂_{M}.

2010 Mathematics Subject Classification. Primary 13F20; Secondary 13L05, 13B25, 13A15, 13G05, 12L10.

Key words and phrases. ultrafilter, integer-valued polynomials, rings of functions, polynomial functions, maximal ideals, prime spectrum, commutative rings, integral domains, products, ultraproducts.

This research was supported by the Austrian Science Fund FWF grant P27816-N26.

In fact, Chabert showed two separate statements independently – both under the assumption that D is Noetherian and one-dimensional and M a maximal ideal of finite index of D:

(1) Every maximal ideal of Int(D) containing Int(D, M) is of the form M_{α} for some α ∈ D̂_{M}.

(2) Every maximal ideal of Int(D) lying over M contains Int(D, M).

For a simplified proof of Chabert’s result, see [4], Lemma 4.4 and the remark following it.

We will show that a modified version of statement (1) holds in far greater generality, for rings of functions. The modification consists in replacing elements of the M-adic completion by ultrafilters.

Whether (2) holds or not for a particular D and a particular subring of D^{E} will have to be examined separately. It is, in some sense, a question of density of the subring in the product ∏_{e∈E} D.

We will work in the following setting:

Definition 1.1. Let D be a commutative ring and E ⊆ D. Let R be a commutative ring and φ: R → ∏_{e∈E} D a monomorphism of rings. φ allows us to interpret the elements of R as functions from E to D.

If all constant functions are contained in φ(R), we call the pair (R, φ) a ring of functions from E to D. We use R = R(E, D) (where φ is understood) to denote a ring of functions from E to D.

Remark 1.2. For our considerations it is vital that R = R(E, D) contain all constant functions, because we will make extensive use of the following fact: when I is an ideal of R = R(E, D), f ∈ I and g ∈ D[x] a polynomial with zero constant term, then g(f) ∈ I, and similarly, if g is a polynomial in several variables over D with zero constant term, and an element of I is substituted for each variable in g, then, an element of I results.

Let us note that considerable research has been done on the spectrum of a power of a ring D^{E} = ∏_{d∈E} D or a product of rings ∏_{e∈E} D_{e}. Gilmer and Heinzer [5, Prop. 2.3] have determined the spectrum of an infinite product of local rings, and Levy, Loustaunau and Shapiro [8] that of an infinite power of ℤ. Our focus here is not on the full product of rings, but on comparatively small subrings and the question of how much information about the spectrum of a ring can be obtained from its embedding in a power of a domain.

One ring can be embedded in different products: Int(D) can be seen as a ring of functions from K to K as well as a ring of functions from D to D. We will glean a lot more information about the spectrum of Int(D) from the second interpretation than from the first.

2. Prime ideals corresponding to ultrafilters

Let R = R(E, D) be a ring of functions from E to D as in Definition 1.1. We will now make precise the concept of ideals corresponding to ultrafilters, and the connection to ultraproducts ∏_{e∈E}^{U}(D/M), where M is a maximal ideal of D, and U an ultrafilter on E. First a quick review of filters, ultrafilters and ultraproducts:

Definition 2.1. Let S be a set. A non-empty collection F of subsets of S is called a filter on S if

(1) ∅ ∉ F.

(2) A, B ∈ F implies A ∩ B ∈ F.

(3) A ⊆ C ⊆ S with A ∈ F implies C ∈ F. A filter F on S is called an ultrafilter on S if, for every C ⊆ S, either C ∈ F or S \ C ∈ F.

Let S be a fixed set and P(S) its power-set. For C ∈ P(S), a superset of C is a set D ∈ P(S) with C ⊆ D ⊆ S. A collection C of subsets of S is said to have the finite intersection property if the intersection of any finitely many members of C is non-empty.

Remark 2.2. Clearly, a necessary and sufficient condition for C ⊆ P(S) to be contained in a filter on S is that C satisfies the finite intersection property. If the finite intersection property is satisfied, then the supersets of finite intersections of members of C form a filter.

Although, strictly speaking, we do not need ultraproducts to prove our results, we will nevertheless introduce them, because they provide context, in particular to Lemma 2.6, and to sections 3 and 5.

Definition 2.3. Let S be an index set and U an ultrafilter on S. Suppose we are given, for each s ∈ S, a ring R_{s}. Then the ultraproduct of rings ∏_{s∈S}^{U} R_{s} is defined as the direct product ∏_{s∈S} R_{s} modulo the congruence relation

(r_{s})_{s∈S} ∼ (t_{s})_{s∈S} ⟺ {s ∈ S | r_{s} = t_{s}} ∈ U.

Ultraproducts of other algebraic structures are defined analogously. The usefulness of ultraproducts is captured by the Theorem of Łoś (cf. [6, Chpt. 3.2] or [7, Prop 1.6.14]) which states that an ultraproduct ∏_{s∈S}^{U} R_{s} satisfies a firstorder formula if and only if the set of indices s for which R_{s} satisfies the formula is in U. Here first-order formula means a formula in the first-order language whose only non-logical symbols (apart from the equality sign) are symbols for the algebraic operations; for instance, + and · in the case of an ultraproduct of rings.

Definition 2.4. Let D be a domain, E ⊆ D, R = R(E, D) a ring of functions, I an ideal of D and F a filter on E.

For f ∈ R(E, D), we let f^{−1}(I) = {e ∈ E | f(e) ∈ I} and define

I_{F} = {f ∈ R(E, D) | f^{−1}(I) ∈ F}

Remark 2.5. Let everything as in Definition 2.4, I, J ideals of D and F, G filters on E. Some easy consequences of Definition 2.4 are:

(1) If I ≠ D then I_{F} ≠ R.

(2) I_{F} is an ideal of R containing R(E, I) = {f ∈ R | f(E) ⊆ I}.

(3) I ⊆ J ⟹ I_{F} ⊆ J_{F}

(4) F ⊆ G ⟹ I_{F} ⊆ I_{G}

Lemma 2.6. Let D be a domain, E ⊆ D, and R = R(E, D) a ring of functions from E to D.

Then for every prime ideal P of D and every ultrafilter U on E, P_{U} is a prime ideal of R.

Proof. Easy direct verification: let fg ∈ P_{U}; because P is a prime ideal of D, the inverse image of P under f · g is the union of f^{−1}(P) and g^{−1}(P). If the union of two sets is in an ultrafilter, then one of them must be in the ultrafilter. Therefore, f ∈ P_{U} or g ∈ P_{U}. Also, P_{U} cannot be all of R because it doesn’t contain the constant function 1. □

One way of looking at P_{U} is by considering the following commuting diagram of ring-homomorphisms, where π and π_{1} mean applying the canonical projection in each factor of the product, and σ and σ_{1} mean factoring through the defining congruence relation of an ultraproduct.

\begin{aligned}R & ∏_{e∈E} D & ∏_{e∈E}^{U} D \\ & ∏_{e∈E}(D/P) & ∏_{e∈E}^{U}(D/P) \end{aligned} φ σ_{1} π π_{1} σ

P_{U} is the kernel of the following composition of ring homomorphisms:

φ: R → ∏_{e∈E} D

followed by the canonical projection

π: ∏_{e∈E} D → ∏_{e∈E}(D/P)

and the canonical projection

σ: ∏_{e∈E}(D/P) → ∏_{e∈E}^{U}(D/P)

Since D/P is an integral domain, any ultraproduct of copies of D/P is also an integral domain, by the Theorem of Łoś. Therefore (0) is a prime ideal of ∏_{e∈E}^{U}(D/P) and hence P_{U} a prime ideal of R. We also see that P_{U} is the inverse image of a prime ideal of ∏_{e∈E} D under φ, and further, of a prime ideal of the ultraproduct ∏_{e∈E}^{U} D under σ_{1} ∘ φ.

3. The set of zero-loci mod M of an ideal of the ring of functions

As before, D is a domain with quotient field K, E ⊆ D and R = R(E, D) a ring of functions from E to D as in Def. 1.1. Especially, recall from Def. 1.1 that R is assumed to contain all constant functions.

Definition 3.1. For M ⊆ D and f ∈ R = R(E, D), let

f^{−1}(M) = {e ∈ E | f(e) ∈ M}.

For an ideal M of D and an ideal I of R, let

Z_{M}(I) = {f^{−1}(M) | f ∈ I}

Recall from Def. 2.4 that for a filter F on E,

M_{F} = {f ∈ R(E, D) | f^{−1}(M) ∈ F}

Remark 3.2. Note that the above definition implies (1) I ⊆ J ⟹ Z_{M}(I) ⊆ Z_{M}(J)

(2) I ⊆ M_{F} ⟺ Z_{M}(I) ⊆ F

Lemma 3.3. Let M be an ideal of D and I an ideal of R. The following are equivalent:

(a) There exists a filter F on E such that I ⊆ M_{F}.

(b) Z_{M}(I) satisfies the finite intersection property.

Proof. If I ⊆ M_{F}, then Z_{M}(I) is contained in F and hence satisfies the finite intersection property. Conversely, if Z_{M}(I) satisfies the finite intersection property then, by Remark 2.2, the supersets of finite intersections of sets in Z_{M}(I) form a filter F on E for which Z_{M}(I) ⊆ F and hence I ⊆ M_{F}. □

In the case where R(E, D) = ∏_{e∈E} D is the ring of all functions from E to D, much more can be said; see the papers by Gilmer and Heinzer [5, Prop. 2.3] (concerning local rings) and Levy, Loustaunau and Shapiro [8] (concerning D = ℤ).

For a field K that is not algebraically closed, we will need, for an arbitrary n ≥ 2, an n-ary form that has no zero but the trivial one. For this purpose, recall how to define a norm form: if L : K is an n-dimensional field extension, multiplication by any w ∈ L is a K-endomorphism ψ_{w} of L. For a fixed choice of a K-basis of L, map every w ∈ L to the determinant of the matrix of ψ_{w} with respect to the chosen basis. This mapping, regarded as a function of the coordinates of w with respect to the chosen basis, is easily seen to be an n-ary form that has no zero but the trivial one.

Lemma 3.4. Let M be a maximal ideal of D such that D/M is not algebraically closed. Then for every ideal I of R = R(E, D), Z_{M}(I) is closed under finite intersections.

Proof. Given f, g ∈ I, we show that there exists h ∈ I with

h^{−1}(M) = f^{−1}(M) ∩ g^{−1}(M).

Consider any finite-dimensional non-trivial field extension of D/M, and let n be the degree of the extension. The norm form of this field extension is a homogeneous polynomial in n ≥ 2 indeterminates whose only zero in (D/M)^{n} is the trivial one. By identifying n − 1 variables, we get a binary form s̄ ∈ (D/M)[x, y] with no zero in (D/M)^{2} other than (0, 0). Let s ∈ D[x, y] be a binary form that reduces to s̄ when the coefficients are taken mod M.

Now, given f and g in I, we set h = s(f, g). By the fact that R contains all constant functions, h is in I. Also, h(e) ∈ M if and only if both f(e) ∈ M and g(e) ∈ M, as desired. □

Lemma 3.5. Let M be a maximal ideal of D and R = R(E, D) a ring of functions such that every f ∈ R takes values in only finitely many residue classes mod M.

Then for every ideal I of R, Z_{M}(I) is closed under finite intersections.

Proof. Again, given f, g ∈ I, we show that there exists h ∈ I with

h^{−1}(M) = f^{−1}(M) ∩ g^{−1}(M).

Let A, B ⊆ D/M be finite sets of residue classes of D mod M such that f(E) is contained in the union of A and g(E) in the union of B.

We can interpolate any function from (D/M)^{2} to (D/M) at any finite set of arguments by a polynomial in (D/M)[x, y]. Pick s̄ ∈ (D/M)[x, y] with s̄(0, 0) = 0 and s̄(a, b) = 1 for all (a, b) ∈ (A × B) \ {(0, 0)}. Let s ∈ D[x, y] be a polynomial with zero constant coefficient that reduces to s̄ when the coefficients are taken mod M.

Now, given f and g in I, we set h = s(f, g). By the fact that R contains all constant functions, h is in I. Also, h(e) ∈ M if and only if both f(e) ∈ M and g(e) ∈ M, as desired. □

Definition 3.6. Let R = R(E, D) be a ring of functions and M an ideal of D. We call f ∈ R an M-unit-valued function if f(e) + M is a unit in D/M for every e ∈ E.

Theorem 3.7. Let M be a maximal ideal of D and I an ideal of R = R(E, D). Assume that either D/M is not algebraically closed or that each function in R takes values in only finitely many residue classes mod M.

(1) I is contained in an ideal of the form M_{F} for some filter F on E if and only if I contains no M-unit-valued function.

(2) Every ideal Q of R that is maximal with respect to not containing any M-unit-valued function is of the form M_{U} for some ultrafilter U on E.

(3) In particular, every maximal ideal of R that does not contain any M-unit-valued function is of the form M_{U} for some ultrafilter U on E.

Proof. Ad (1). If I is contained in an ideal of the form M_{F}, I cannot contain any M-unit-valued function, because F doesn’t contain the empty set.

Conversely, suppose that I does not contain any M-unit-valued function. Then ∅ ∉ Z_{M}(I). By Lemmata 3.4 and 3.5, Z_{M}(I) is closed under finite intersections. Z_{M}(I), therefore, satisfies the finite intersection property. By Remark 2.2, Z_{M}(I) is contained in a filter F on E. For this filter, I ⊆ M_{F}, by Remark 3.2.

Ad (2). Suppose Q is maximal with respect to not containing any M-unit-valued function. By (1), Q ⊆ M_{F} for some filter F. Refine F to an ultrafilter U. Then, by Remark 2.5, Q ⊆ M_{F} ⊆ M_{U}, and M_{U} doesn’t contain any M-unit-valued function. Since Q is maximal with this property, Q = M_{U}.

(3) is a special case of (2). □

4. A dichotomy of maximal ideals

In what follows, D is always a domain with quotient field K, E ⊆ D and R = R(E, D) a ring of functions from E to D as in Def. 1.1. When the interpretation of R as a subring of ∏_{e∈E} D is understood, then for M ⊆ D we let

R(E, M) = {f ∈ R | f(E) ⊆ M}.

Proposition 4.1. Let M be a maximal ideal of D and Q a maximal ideal of R = R(E, D). Then exactly one of the following two statements holds:

(1) Q contains R(E, M) = {f ∈ R | f(E) ⊆ M}

(2) Q contains an element f with f(e) ≡ 1 mod M for all e ∈ E.

Proof. The two cases are mutually exclusive, because any ideal Q satisfying both statements must contain 1.

Now suppose Q does not contain R(E, M). Let g ∈ R(E, M) \ Q. By the maximality of Q, 1 = h(x)g(x) + f(x) for some h ∈ R and f ∈ Q. We see that f(x) = 1 − h(x)g(x) ∈ Q satisfies f(e) ≡ 1 mod M for all e ∈ E. □

Recall that a function f ∈ R is called M-unit-valued if f(e) + M is a unit in D/M for every e ∈ E.

Lemma 4.2. Let M be an ideal of D and Q an ideal of R = R(E, D). The following are equivalent:

(A) Q contains an element f with f(e) ≡ 1 mod M for all e ∈ E.

(B) Q contains an M-unit-valued function that takes values in only finitely many residue classes mod M.

Proof. To see that the a priori weaker statement implies the stronger, let g ∈ Q be an M-unit-valued function taking only finitely many different values mod M. Let d_{1}, …, d_{k} ∈ D be representatives of the finitely many residue classes mod M intersecting g(E) non-trivially, and u ∈ D an inverse mod M of (−1)^{k+1}d_{1} · . . . · d_{k}.

Then

h(x) = ∏_{i=1}^{k}(g(x) − d_{i}) − (−1)^{k}d_{1} · . . . · d_{k}

is in Q and h(e) ≡ (−1)^{k+1}d_{1} · . . . · d_{k} mod M for all e ∈ E. Therefore f(x) = uh(x) ∈ Q satisfies f(e) ≡ 1 mod M for all e ∈ E. □

Proposition 4.3. Let M be a maximal ideal of D and Q a maximal ideal of R = R(E, D). If each f ∈ R takes values in only finitely many residue classes mod M (in particular, if D/M happens to be finite) then exactly one of the following statements holds:

(1) Q contains R(E, M) = {f ∈ R | f(E) ⊆ M}

(2) Q contains an M-unit-valued function.

Proof. This follows directly from Proposition 4.1 and Lemma 4.2. □

The Propositions in this section partition the maximal ideals of R lying over a maximal ideal M of D into two types: those containing R(E, M) (the kernel of the restriction to R of the canonical projection π: ∏_{e∈E} D ⟶ ∏_{e∈E}(D/M)), and the others.

In some cases, it is known that all maximal ideals of R lying over M contain R(E, M), notably if R = Int(D) and M is finitely generated and of finite index in D [1, Ch. V], [4, Lemma 4.4]. We will find a sufficient condition for all maximal ideals of R lying over M to contain R(E, M) in Theorem 6.4.

We must not discount the possibility of a maximal ideal Q lying over M containing an M-unit-valued function, however. If D is an infinite domain, D[x] is embedded in D^{D} by mapping every polynomial to the corresponding polynomial function. When D/M is not algebraically closed, then there are certainly maximal ideals of D[x] lying over M that contain polynomials without a zero mod M.

5. Prime ideals containing R(E, M)

We are now in a position to characterize the prime ideals of R containing R(E, D) as being precisely the ideals of the form M_{U} for ultrafilters U on E, under the following hypothesis: every f ∈ R takes values in only finitely many residue classes of M.

This hypothesis may seem only marginally weaker than the assumption that D/M is finite. Note however, that it is sometimes satisfied for infinite D/M under perfectly natural circumstances, for instance, when E intersects only finitely many residue classes of M^{n} for each n ∈ ℕ (E precompact), and R consists of functions that are uniformly M-adically continuous.

As in the case of integer-valued polynomials, we can show that every prime ideal of R(E, D) containing R(E, M) is maximal under certain conditions, notably if D/M is finite. The proof for Int(D), when D/M is finite [1, Lemma V.1.9.], carries over practically without change. Note that Definition 1.1 ensures that every ring of functions R contains all constant functions – an essential requirement of the following proof.

Lemma 5.1. Let M be a maximal ideal of D such that every function in R = R(E, D) takes values in only finitely many residue classes mod M, and Q a prime ideal of R(E, D) containing R(E, M). Then Q is maximal and R/Q is isomorphic to D/M.

Proof. Let Q be a prime ideal of R(E, D) containing R(E, M), and A a system of representatives of D mod M. It suffices to show that A (viewed as a set of constant functions) is also a system of representatives of R mod Q. Let f ∈ R(E, D) and a_{1}, …, a_{r} ∈ A the representatives of those residue classes of M that intersect f(E) non-trivially. Then ∏_{i=1}^{r}(f − a_{i}) is in R(E, M) ⊆ Q and, Q being prime, one of the factors (f − a_{i}) must be in Q. This shows that f is congruent mod Q to one of the constant functions a_{1}, …, a_{r}, and, in particular, to an element of A. Therefore, A is a system of representatives of R(E, D) mod Q. □

Lemma 5.2. Let R = R(E, D) a ring of functions and M a maximal ideal of D such that every f ∈ R takes values in only finitely many residue classes of M. Let I be an ideal of R.

Then I is contained in an ideal of the form M_{F} for a filter F on E if and only if R(E, M) ⊆ I.

Proof. R(E, M) ⊆ I is equivalent to I not containing an M-unit-valued function, by Proposition 4.3. The statement therefore follows from part (1) of Theorem 3.7.

□

Theorem 5.3. Let R = R(E, D) a ring of functions, and M a maximal ideal of D. If every f ∈ R takes values in only finitely many residue classes of M (and, in particular, if D/M is finite), then the prime ideals of R containing R(E, M) are exactly the ideals of the form M_{U} with U an ultrafilter on E. Each of them is maximal and its residue field isomorphic to D/M.

Proof. Let Q be a prime ideal of R containing R(E, M). By Lemma 5.1, Q is maximal and R/Q is isomorphic to D/M. By Lemma 5.2, Q ⊆ M_{F} for some filter F on E. F can be refined to an ultrafilter U on E, and then Q ⊆ M_{F} ⊆ M_{U} ≠ R, by Remark 2.5. Since Q is maximal, Q = M_{U} follows.

Conversely, every ideal of the form M_{U} for an ultrafilter U on E is prime, by Lemma 2.6, and contains R(E, M), by Remark 2.5. □

Note, in particular, that Theorems 3.7 and 5.3 apply to R = Int(E, D). In this way, we see, when M is a maximal ideal of finite index in D, that prime ideals of Int(E, D) containing Int(D, M) are inverse images of prime ideals of D^{E}, and ultimately come from ultrapowers of (D/M), as in the discussion after Lemma 2.6.

6. Divisible rings of functions

Let R ⊆ D^{E} be a ring of functions and M a maximal ideal of D. We have seen that we can describe those maximal ideals of R lying over M that contain R(E, M). We would like to know under what conditions this holds for every maximal ideal of R lying over M.

In the case where M is a maximal ideal of finite index in a one-dimensional Noetherian domain D, Chabert showed that every maximal ideal of Int(D) lying over M contains Int(D, M), cf. [1, Prop. V.1.11] and [4, Lemma 3.3]. Once we know this, Theorem 5.3 is applicable. It can be used to give an alternative proof of the fact that every prime ideal of Int(D) lying over M is maximal and of the form M_{α} = {f ∈ Int(D) | f(α) ∈ M̂} for an element α in the M-adic completion of D.

We will now generalize Chabert’s argument from integer-valued polynomials to a class of rings of functions which we call divisible. Note that we do not have to restrict ourselves to Noetherian domains; we only require the individual maximal ideal for which we study the primes of R lying over it to be finitely generated. It is true that our questions only localize well when the domain is Noetherian, but we will pursue a different course, not relying on localization. Definition 6.1. Let R be a commutative ring and E ⊆ R. We call a ring of functions R ⊆ R^{E} divisible if it has the following property: If f ∈ R is such that f(E) ⊆ cR for some non-zero c ∈ R, then every function g ∈ R^{E} satisfying cg(x) = f(x) is also in R.

We call R weakly divisible if for every f ∈ R and every non-zero c ∈ R such that f(E) ⊆ cR, there exists a function g ∈ R with cg(x) = f(x).

If R is a domain, we note that g(x) in the above definition is unique and that, therefore, for domains, weakly divisible is equivalent to divisible. Example 6.2. (1) Int(E, D) is divisible. - This is our motivation.

(2) If D is a valuation domain with maximal ideal M then the ring of uniformly M-adically continuous functions from E to D (E ⊆ D equipped with subspace topology of M-adic topology) is a divisible ring of functions.

We now consider minimal prime ideals of non-zero principal ideals, that is, P containing some p ≠ 0 such that there is no prime ideal strictly contained in P and containing p. If D is Noetherian, this condition reduces to “ht(P) = 1”. In non-Noetherian domains, we find examples with ht(P) > 1, for instance, the maximal ideal of a finite-dimensional valuation domain. Lemma 6.3. Let R be a domain, P a finitely generated prime ideal that is a minimal prime of a non-zero principal ideal (p) ⊆ P. Then there exist m ∈ ℕ and s ∈ R \ P such that sP^{m} ⊆ pR. Proof. In the localization R_{P}, P_{P} is the radical of pR_{P}. Therefore, since P (and hence P_{P}) is finitely generated, there exists m ∈ ℕ with P_{P}^{m} ⊆ pR_{P} and in particular P^{m} ⊆ pR_{P}. The ideal P^{m} is also finitely generated, by p_{1}, …, p_{k}, say. Let a_{i} ∈ R_{P} with p_{i} = pa_{i}. By considering the fractions a_{i} = r_{i}/s_{i} (with r_{i} ∈ R and s_{i} ∈ R \ P), and setting s = s_{1} · . . . · s_{k}, we see that sP^{m} ⊆ pR as desired. □

Theorem 6.4. Let D be a domain and P a finitely generated prime ideal that is a minimal prime of a non-zero principal ideal. Let R ⊆ D^{E} be a divisible ring of functions from E to D. Then every prime ideal Q of R with Q ∩ D = P contains R(E, P).

Proof. Let f ∈ R(E, P). Let p ∈ P non-zero and such that there is no prime ideal P_{1} with (p) ⊆ P_{1} ⊊ P. By Lemma 6.3, there are m ∈ ℕ and s ∈ D \ P such that sP^{m} ⊆ pD. Then sf^{m} ∈ R(E, pD). Since R is divisible, sf^{m} = pg for some g ∈ R(E, D). Therefore, sf^{m} ∈ p R(E, D) ⊆ Q. As Q is prime and s ∉ Q, we conclude that f ∈ Q. □

Corollary 6.5. Let D be a domain, M a finitely generated maximal ideal of height 1, and E a subset of D. Let R ⊆ D^{E} be a divisible ring of functions from E to D, such that each f ∈ R takes its values in only finitely many residue classes of M in D.

Then the prime ideals of R lying over M are precisely the ideals of the form M_{U} for an ultrafilter U on E. Each M_{U} is a maximal ideal and its residue field isomorphic to D/M.

Proof. This follows from Theorem 6.4 via Theorem 5.3. □

To summarize, we can, using ultrafilters, describe certain prime ideals of a ring of functions R = R(E, D) lying over a maximal ideal M pretty well: namely, those prime ideals that do not contain M-unit-valued functions (Theorem 3.7), or that contain R(E, M) (Theorem 5.3).

We have, so far, little information about when all prime ideals of R lying over M are of this form, apart from the sufficient condition in Theorem 6.4.

If we restrict our attention to rings of functions R with D[x] ⊆ R(E, D) ⊆ D^{E}, it would be interesting to find a precise criterion, perhaps involving topological density, for this property.

Note that in the “nicest” case, that of Int(D), where D is a Dedekind ring with finite residue fields, not only is Int(D, M) contained in every prime ideal of Int(D) lying over a maximal ideal M of D, but also Int(D) is dense in D^{D} with product topology of discrete topology on D [2,3].

References

[1] P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, vol. 48 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.

[2] P.-J. Cahen, J.-L. Chabert, and S. Frisch, Interpolation domains, J. Algebra 225 (2000), 794–803.

[3] S. Frisch, Interpolation by integer-valued polynomials, J. Algebra 211 (1999), 562–577.

[4] S. Frisch, Integer-valued polynomials on algebras, J. Algebra 373 (2013), 414–425, Corrigendum: 412 (2014) p282.

[5] R. Gilmer and W. Heinzer, Imbeddability of a commutative ring in a finite-dimensional ring, Manuscripta Math. 84 (1994), 401–414.

[6] M. Goldstern and H. Judah, The incompleteness phenomenon, A K Peters, Ltd., Wellesley, MA, 1995. A new course in mathematical logic, With a foreword by Saharon Shelah.

[7] P. G. Hinman, Fundamentals of mathematical logic, A K Peters, Ltd., Wellesley, MA, 2005.

[8] R. Levy, P. Loustaunau, and J. Shapiro, The prime spectrum of an infinite product of copies of Z, Fund. Math. 138 (1991), 155–164.

Department of Analysis and Number Theory (5010), Technische Universität Graz, Kopernikusgasse 24, 8010 Graz, Austria

  E-mail address: frisch@math.tugraz.at