J. Number Theory 56 (1996) 396–403

Substitution and Closure of Sets under Integer-Valued Polynomials

Sophie Frisch

Abstract.

Let R be a domain and K its quotient-field. For a subset S of K, let F_{R}(S) be the set of polynomials f ∈ K[x] with f(S) ⊆ R and define the R-closure of S as the set of those t ∈ K for which f(t) ∈ R for all f ∈ F_{R}(S). The concept of R-closure was introduced by McQuillan (J. Number Theory 39 (1991), 245–250), who gave a description in terms of closure in P -adic topology, when R is a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker than P -adic topology, which allows us to treat ideals of infinite index, and derive a characterization of R-closure when R is a Krull ring. This gives us a criterion for F_{R}(S) = F_{R}(T), where S and T are subsets of K. As a corollary we get a generalization to Krull rings of R. Gilmer’s result (J. Number Theory 33 (1989), 95–100) characterizing those subsets S of a Dedekind ring with finite residue fields for which F_{R}(S) = F_{R}(R).

1. Introduction.

Let R be a domain and K its quotient-field. The ring of integer-valued polynomials on R consists of those polynomials in K[x] that map R to itself, when acting as a function on K by substitution of the variable. (The name stems from the classical case, where R is the ring of integers in a number field.) Although this ring has been the object of extensive study (originating with two seminal papers by Pólya [5] and Ostrowski [4]), some natural questions have not been considered until fairly recently. If, for a subset S of K, we denote by F_{R}(S) the set of R-valued polynomials on S, F_{R}(S) = {f ∈ K[x] | f(S) ⊆ R}, a question one may ask is which subsets of R can be substituted for R to define the ring of integer-valued polynomials, F_{R}(S) = F_{R}(R). R. Gilmer [1] characterized those subsets for a Dedekind ring with finite residue fields.

To investigate when F_{R}(S) = F_{R}(T), for arbitrary S, T ⊆ K, D. L. McQuillan [3] introduced the R-closure of a set, R−cl(S) = {t ∈ K | ∀f ∈ F_{R}(S) f(t) ∈ R}. Clearly, F_{R}(S) ⊆ F_{R}(T) if and only if T ⊆ R−cl(S). For a Dedekind ring with finite residue fields McQuillan gave a description of the R-closure in terms of the closures in P -adic topology, where P runs through the maximal ideals of R.

integer-valued polynomial closure

In this paper we introduce a topology related to, but weaker than P -adic topology, which allows us to handle prime ideals of infinite index. When R is a Dedekind ring (or more generally a Krull ring), we give a characterization of the R-closure of sets in terms of “weak P -adic topology.” As a corollary we get a generalization of Gilmer’s result to Krull rings.

2. Weak ℐ-adic topology.

All rings considered will be commutative with identity. A descending chain of ideals in a ring R is understood to be a sequence ℐ = {I_{n} | n ∈ ℕ} of ideals with I_{n+1} ⊆ I_{n} and we set I_{0} := R. (The natural numbers ℕ do not contain 0, but ℕ_{0} = ℕ ∪ {0}.)

Definition. Let R be a ring, ℐ a descending chain of ideals in R and M an R-module. We define weak ℐ-adic topology on M by giving a neighborhood basis for m ∈ M: U(m) = ⋃_{n=1}^{∞} U_{n}(m), where U_{n}(m) consists of all sets M \ ⋃_{j=1}^{n} E_{j}, such that each E_{j} is contained in m + I_{j−1}M and is a finite union of residue classes of I_{j}M, other than m + I_{j}M, in M. (Thus U_{n}(m) = {m + U | U ∈ U_{n}(0)}.)

Definition. If I is an ideal of R, weak I -adic topology is defined as weak ℐ-adic topology for ℐ = {I^{n} | n ∈ ℕ}.

To see that the neighborhood bases U(m) define a topology on M we check that (1) ∀ U ∈ U_{n}(m) m + I_{n}M ⊆ U, in particular, m ∈ U,

(2) U, V ∈ U_{n}(m) ⟹ U ∩ V ∈ U_{n}(m),

(3) ∀z ∈ U ∈ U_{n}(m) ∃V ∈ U_{n}(z) V ⊆ U.

Ad (3): If z ≡ m mod I_{n}M then U_{n}(z) = U_{n}(m). If l < n is maximal such that z ≡ m mod I_{l}M and U = M \ ⋃_{j=1}^{n} E_{j}, each E_{j} being a finite union of residue classes other than m + I_{j}M in m + I_{j−1}M, then V = M \ (⋃_{j=1}^{l+1} E_{j} ∪ (m + I_{l+1}M)) ⊆ U and V ∈ U_{l+1}(z) ⊆ U_{n}(z).

Remarks. (3) shows that basis neighborhoods are open and (1) implies that weak I -adic topology is actually weaker than I -adic topology.

Perhaps a more natural way to look at weak ℐ-adic topology on a ring R is the following: If ⋂_{n=1}^{∞} I_{n} = (0) then there is an embedding ι of R (otherwise of R / ⋂_{n=1}^{∞} I_{n}) into ∏_{n=1}^{∞} I_{n−1}/I_{n} (for n ∈ ℕ let {c_{j}^{(n)} | 0 ≤ j < [I_{n−1} : I_{n}]} be a residue system of I_{n−1} mod I_{n} and for r ∈ R define ι(r) := (c_{j_{n}(r)}^{(n)})_{n=1}^{∞} by r ≡ ∑_{n=1}^{N} c_{j_{n}(r)}^{(n)} mod I_{N} for all N ∈ ℕ). Weak ℐ-adic topology is then induced on R by the product topology of co-finite topology on each factor I_{n−1}/I_{n}. If, for an ideal I in R, we compare I -adic topology to weak I -adic topology, we see that the former is induced by product topology of discrete topology on ∏_{n=1}^{∞} I^{n−1}/I^{n}, and thus is stronger than the latter, and that equality holds if and only if [I^{n−1} : I^{n}] is finite for all n ∈ ℕ.

3. Local investigations.

Throughout the “local” section, v is a discrete valuation (with value group equal to ℤ and v(0) = ∞) on a field K and R_{v} its valuation ring with maximal ideal M_{v}. If S is a set contained in R_{v} we denote the closure of S in weak M_{v} -adic topology by \overline{S}. We shall see that weak M_{v} -adic topology on R_{v} arises naturally as “topology of closure under integer-valued polynomials,” in that \overline{S} = R_{v}−cl(S). We need a few technical Lemmata.

Lemma 1. Let R be a subring of a ring R′, and ℐ, 𝒥 descending chains of ideals in R and R′, respectively. If there exists a strictly increasing function φ : ℕ → ℕ with φ(1) = 1 such that for all n ∈ ℕ, I_{n} = J_{k} ∩ R whenever φ(n) ≤ k < φ(n + 1), then weak ℐ-adic topology on R is equal to the topology inherited from weak 𝒥-adic topology .

Proof. Fix t ∈ R. If C is a residue class of J_{k} in R′ then either R ∩ C = ∅ or C = r + J_{k} and C ∩ R = r + (J_{k} ∩ R) for some r ∈ R. Moreover, if C = r + J_{k} with r ∈ R such that r ≡ t (J_{k−1}), but r ≢ t (J_{k}), then we must have J_{k−1} ∩ R ≠ J_{k} ∩ R, so there exists n ∈ ℕ with k = φ(n); and if we put D = C ∩ R then D = r + I_{n} ≠ t + I_{n} and D ⊆ t + I_{n−1} . Conversely, if for some r ∈ R, D = r + I_{n} with r ≡ t (I_{n−1}) and r ≢ t (I_{n}) then D = R ∩ C, where C = r + J_{φ(n)} ⊆ t + J_{φ(n)−1} and C ≠ t + J_{φ(n)} . It follows immediately from these considerations that the intersections of weak 𝒥-adic basis neighborhoods of t with R are precisely the weak ℐ-adic basis neighborhoods of t. □

Remark: If v′ is an extension of the discrete valuation v to a finite-dimensional extension K′ of K then Lemma 1 implies equality of weak M_{v} -adic topology on R_{v} with the topology inherited from weak M_{v′} -adic topology. Namely, if e ∈ ℕ is the index of the valuation group of v in the valuation group of v′ then M_{v}^{n} = M_{v′}^{k} ∩ R whenever e · (n − 1) + 1 ≤ k < e · n + 1.

Lemma 2. Let f ∈ R_{v}[x], not all of whose coefficients lie in M_{v}, split over K, as f(x) = d(x − b_{1}) · . . . · (x − b_{m}) · (x − c_{1}) · . . . · (x − c_{l}), where v(b_{i}) < 0 and v(c_{i}) ≥ 0, and put f_{+}(x) = (x − c_{1}) · . . . · (x − c_{l}). Then v(f(r)) = v(f_{+}(r)) for all r ∈ R_{v}.

Proof. ∀r ∈ R_{v} v(r − b_{i}) = v(b_{i}) and v(f(r)) = v(d) + ∑_{i=1}^{m} v(b_{i}) + v(f_{+}(r)); we show v(d) = − ∑_{j=1}^{m} v(b_{i}). Consider d^{−1}f(x) = x^{n} + a_{n−1}x^{n−1} + … + a_{0} . Since f ∈ R_{v}[x] \ M_{v}[x], v(d) = − min_{0≤i≤n} v(a_{i}). But the a_{i} are the elementary symmetric polynomials in the b_{i} and c_{i}, so min_{0≤i≤n} v(a_{i}) = v(a_{n−m}) = ∑_{i=1}^{m} v(b_{i}). □

Lemma 3. Let S be a set contained in R_{v} and a ∈ R_{v}. Then

a ∈\overline{S} ⟹ ∀f ∈ K[x] ∃s ∈ S v(f(s)) ≤ v(f(a)) .

Proof. If S = ∅ or f is constant or f(a) = 0 the statement is trivial; from now on, assume S ≠ ∅, deg(f) ≥ 1, and f(a) ≠ 0. First consider a monic f ∈ R_{v}[x] that splits over K: f(x) = ∏_{i=1}^{n}(x−c_{i}) with c_{i} ∈ R_{v}. Since f(a) ≠ 0, l = max_{i} v(a−c_{i}) exists, and v(f(a)) = ∑_{i=1}^{n} v(a − c_{i}) = ∑_{j≥1} |{i | a ≡ c_{i} mod M_{v}^{j}}| = ∑_{j=1}^{l} |{i | a ≡ c_{i} mod M_{v}^{j}}|.

Since a ∈ \overline{S}, and S therefore intersects every U ∈ U_{l+1}(a), either there exists s_{0} ∈ S ∩ (a + M_{v}^{l+1}), or there exists m ≤ l such that S intersects infinitely many residue classes of M_{v}^{m+1} in a + M_{v}^{m}. In the first case, v(f(s_{0})) = ∑_{j=1}^{l}|{i | s_{0} ≡ c_{i} mod M_{v}^{j}}| = ∑_{j=1}^{l}|{i | a ≡ c_{i} mod M_{v}^{j}}| = v(f(a)). In the second case, pick t_{0} ∈ S ∩ (a + M_{v}^{m}) such that t_{0} ≢ c_{i} mod M_{v}^{m+1} for i = 1, …, n then v(f(t_{0})) = ∑_{j=1}^{m}|{i | t_{0} ≡ c_{i} mod M_{v}^{j}}| = ∑_{j=1}^{m}|{i | a ≡ c_{i} mod M_{v}^{j}}| ≤ v(f(a)).

Now for a general f ∈ K[x] (with deg(f) ≥ 1 and f(a) ≠ 0), write f as c · g with c ∈ K, g ∈ R_{v}[x]\M_{v}[x]. It suffices to prove the claim for g. Let K′ be the splitting field of g over K, v′ an extension of v to K′ (normalized to have value group ℤ, such that on K, we have v′ = e·v, e ∈ ℕ). Over K′ we get g(x) = d(x−c_{1}) … (x−c_{n})(x−b_{1}) … (x−b_{m}) with v′(c_{i}) ≥ 0, v′(b_{i}) < 0. By Lemma 2, for every t ∈ R_{v′}, v′(g(t)) = v′(g_{+}(t)), where g_{+}(x) = (x−c_{1}) … (x−c_{n}). But now we know there exists s ∈ S with v′(g_{+}(s)) ≤ v′(g_{+}(a)) (using the fact that the closure of S in weak M_{v} -adic topology is contained in the closure with respect to weak M_{v′} -adic topology); and v(g(s)) = e^{−1}v′(g(s)) = e^{−1}v′(g_{+}(s)) ≤ e^{−1}v′(g_{+}(a)) = e^{−1}v′(g(a)) = v(g(a)). □

Lemma 4. Let S be a set contained in R_{v} and a ∈ R_{v}. Then

a ∉\overline{S} ⟹ ∃f ∈ [S][x] ∀s ∈ S v(f(s)) > v(f(a)) ,

where [S] denotes the ring generated by S in K.

Proof. If S = ∅ the statement is trivial, so assume S ≠ ∅. Since a ∉\overline{S}, there exists a basis-neighborhood of a which S doesn’t intersect, and hence a minimal N ∈ ℕ such that S ∩ (a + M_{v}^{N}) = ∅ and S meets only finitely many residue classes of M_{v}^{n} in a + M_{v}^{n−1} for all n ≤ N. Inductively, from k = N − 1 down to k = 0, we construct a sequence of polynomials f_{k} ∈ [S][x] such that v(f_{k}(s)) > v(f_{k}(a)) for all s ∈ S ∩ (a + M_{v}^{k}).

Define f_{N−1}(x) = ∏_{i=1}^{m}(x − s_{i}), where s_{1}, …, s_{m} ∈ S are representatives of the different residue classes of M_{v}^{N} that S intersects in a + M_{v}^{N−1} . (Minimality of N and the fact that S ≠ ∅ guarantee that S intersects a + M_{v}^{N−1} ; hence m ≠ 0.) Then v(f_{N−1}(s)) ≥ m(N − 1) + 1 > m(N − 1) = v(f_{N−1}(a)) for all s ∈ S ∩ (a + M_{v}^{N−1}).

Given f_{k} such that for all s ∈ S ∩ (a + M_{v}^{k}) v(f_{k}(s)) ≥ c while v(f_{k}(a)) = c − 1, we construct f_{k−1} . Set d = min{v(f_{k}(s)) | s ∈ S ∩ (a + M_{v}^{k−1})}. If d ≥ c then f_{k−1} = f_{k} works. If d < c, let t_{1}, …, t_{l} ∈ S be representatives of the different residue classes of M_{v}^{k} in a + M_{v}^{k−1} , other than a + M_{v}^{k}, that S intersects. Define g(x) = ∏_{i=1}^{l}(x − t_{i}) and f_{k−1} = g^{c−d} · f_{k}. Putting together the facts that

∀s ∈ S ∩ ((a + M_{v}^{k−1}) \ (a + M_{v}^{k})) v(g(s)) ≥ l(k − 1) + 1 and v(f_{k}(s)) ≥ d ,
∀t ∈ a + M_{v}^{k} v(g(t)) = l(k − 1) ,
and ∀s ∈ S ∩ (a + M_{v}^{k}) v(f_{k}(s)) ≥ c, while v(f_{k}(a)) = c − 1 ,

we see that v(f_{k−1}(a)) = (c − d)l(k − 1) + c − 1, while v(f_{k−1}(s)) ≥ (c − d)l(k − 1) + c for all s ∈ S ∩ a + M_{v}^{k−1} . □

Proposition 1. If A and S are sets contained in R_{v} then

F_{R_{v}}(S) ⊆ F_{R_{v}}(A) ⟺ A ⊆\overline{S} .

Proof. For any a ∈\overline{S} Lemma 3 shows that F_{R_{v}}(S) ⊆ F_{R_{v}}(\{a\}). Conversely, if a ∉\overline{S}, Lemma 4 allows us to construct a member of F_{R_{v}}(S) \ F_{R_{v}}({a}) by multiplying the f in the Lemma by a constant c ∈ K with v(c) = − min_{s∈S} v(f(s)). The statement for A now follows from the fact that F_{R_{v}}(A) = ⋂_{a∈A} F_{R_{v}}({a}). □

Corollary. If A and S are sets contained in R_{v} then

(i) F_{R_{v}}(S) = F_{R_{v}}(A) ⟺ \overline{A} =\overline{S} (ii) R_{v}−cl(S) =\overline{S} .

4. Results for Krull-rings.

From now on, let R be a Krull ring, K its field of fractions, and 𝒫 the set of height 1 prime ideals of R. If P ∈ 𝒫, we denote by P^{(n)} , n ∈ ℕ, the symbolic powers of P, P^{(n)} = (P_{P})^{n} ∩ R, where P_{P} is the extension of P to the localization R_{P} . By \overline{S} we now mean the closure of S in the specified topology, be it weak {P^{(n)} | n ∈ ℕ}-adic, weak P -adic or P -adic. A subset S of K is called R-fractional if S ⊆ d^{−1}R for some d ∈ R. As with Dedekind rings with finite residue fields (McQuillan [3]), the case of non-R-fractional sets is simple (I thank F. Halter-Koch for spiffying up the following proposition, which I had only shown for Krull rings, and in a more pedestrian manner.).

Proposition 2. Let R be an integrally closed domain with quotient field K. If A ⊆ K is not R-fractional then F_{R}(A) consists only of the constant polynomials with values in R and hence R−cl(A) = K.

Proof. Suppose f ∈ F_{R}(A), deg f = n > 0. There exists c ≠ 0 in R such that cf = g ∈ R[x], g(x) = c_{n}x^{n} + … + c_{0} , c_{n} ≠ 0. For every a ∈ A, g(a) ∈ R implies that c_{n}a is integral over R, and therefore c_{n}a ∈ R. Thus A ⊆ c_{n}^{−1}R. □

We now turn to R-fractional sets.

Theorem 1. Let A and B be subsets of d^{−1}R, d ∈ R, then

(i) F_{R}(A) ⊆ F_{R}(B) ⟺ ∀P ∈ 𝒫 B ⊆ \overline{A} in weak \{P^{(n)}\}−adic topology on d^{−1}R

(ii) R−cl(A) is the intersection of all weak {P^{(n)}} − adic closures of A, P ∈ 𝒫.

Proof. In the case where A, B ⊆ R , we show that the following are equivalent:

(1) F_{R}(A) ⊆ F_{R}(B)

(2) ∀P ∈ 𝒫, B ⊆\overline{A} in weak P_{P} −adic topology on R_{P}

(3) ∀P ∈ 𝒫, B ⊆\overline{A} in weak \{P^{(n)}\}−adic topology on R .

(1 ⇒ 2) Suppose B ⊈ \overline{A} in weak P_{P} -adic topology for some fixed P ∈ 𝒫. Then by Lemma 4 there exists a polynomial f ∈ [A][x] and an integer n, such that for all a ∈ A v_{P} (f(a)) ≥ n, and for some b ∈ B v_{P} (f(b)) < n. By the Approximation Theorem for Krull-rings [2, p90], there is a c ∈ K with v_{P} (c) = −n and v_{Q}(c) ≥ 0 for all Q ≠ P, Q ∈ 𝒫. Then c · f ∈ F_{R_{P}} (A), but c · f ∉ F_{R_{P}} (B). Also, for Q ≠ P, Q ∈ 𝒫, c · f ∈ R_{Q}[x] ⊆ F_{R_{Q}}(A). Therefore, c · f is in F_{R}(A), but not in F_{R}(B).

(2 ⇒ 1) By Proposition 1, B ⊆ \overline{A} in weak P_{P} -adic topology implies F_{R_{P}} (A) ⊆ F_{R_{P}} (B). Using R =⋂_{P ∈𝒫} R_{P} we get F_{R}(A) =⋂_{P ∈𝒫} F_{R_{P}} (A) ⊆⋂_{P ∈𝒫} F_{R_{P}} (B) = F_{R}(B).

(2 ⇔ 3) Weak {P^{(n)}}-adic topology on R is – by definition of P^{(n)} and Lemma 1 – exactly what R inherits from weak P_{P} -adic topology on R_{P} .

To reduce the fractional sets case to the subsets of R case we convince ourselves that:

(4) F_{R}(A) ⊆ F_{R}(B) if and only if F_{R}(dA) ⊆ F_{R}(dB) and

(5) For every P ∈ 𝒫, B ⊆ \overline{A} in weak \{P^{(n)}\}-adic topology on d^{−1}R if and only if dB ⊆ \overline{dA} in weak \{P^{(n)}\}-adic topology on R.

Ad (4) Consider φ_{d}: K[x] → K[x], φ_{d}(f(x)) = f(d^{−1}x) . Clearly, φ_{d}(F_{R}(S)) = F_{R}(dS) for any set S ⊆ K. Because φ_{d} is a permutation of K[x], φ_{d}(S) ⊆ φ_{d}(T) if and only if S ⊆ T for all S, T ⊆ K.

Ad (5) ψ: d^{−1}R → R, ψ(x) = dx (as an R-module isomorphism) is a homeomorphism between the ℐ-adic topologies on d^{−1}R and R for any descending sequence of ideals ℐ.

The characterization of R−cl(A) is now an easy consequence of its definition as the unique largest set B with F_{R}(A) ⊆ F_{R}(B). □

In what follows, we use the fact that P^{(n)} = P^{n} whenever P^{n} is a primary ideal. This is always the case if P is a maximal ideal, but also when P is a principal prime ideal in a unique factorization domain; so that in these cases, weak {P^{(n)}}-adic topology is just weak P -adic topology. Also note that weak I -adic topology is equal to I -adic topology whenever [R : I^{n}] is finite for all n, such that for a height 1 prime ideal P of finite index in a Krull ring, weak {P^{(n)}}-adic topology is simply P -adic topology. In the case of a Dedekind ring with finite residue fields, the following result is due to McQuillan [3].

Corollary. Let (R, 𝒫) be a Dedekind ring and its set of maximal ideals or a UFD and its set of principal prime ideals. If A and B are subsets of d^{−1}R, d ∈ R, then

(i) F_{R}(A) ⊆ F_{R}(B) ⟺ ∀P ∈ 𝒫 B ⊆\overline{A} in weak P −adic topology on d^{−1}R (ii) R−cl(A) is the intersection of all weak P -adic closures of A, P ∈ 𝒫.

Theorem 2. Let S be a set contained in a subring A of a Krull ring R. Then F_{R}(S) = F_{R}(A) if and only if for every height 1 prime ideal P of R

(a) for all n ∈ ℕ with [A : P^{(n)} ∩ A] finite, S contains a complete system of residues of P^{(n)} ∩ A in A and

(b) for the minimal N (if such exists) with [A : P^{(N)} ∩ A] infinite, S intersects infinitely many residue classes of P^{(N)} ∩ A in every residue class of P^{(N−1)} ∩ A in A.

Proof. The condition is clearly necessary and sufficient for S to intersect every weak {P^{(n)}}-adic neighborhood for all P ∈ 𝒫 of every a ∈ A, that is for A to be contained in the closure of S in weak {P^{(n)}}-adic topology for all P ∈ 𝒫. □

Corollary 1. If S is a subset of a Krull ring R then F_{R}(S) = F_{R}(R) if and only if S contains a complete residue system of P^{n} in R for every n ∈ ℕ for every finite index P ∈ 𝒫 and infinitely many elements incongruent mod P for every P ∈ 𝒫 of infinite index.

Proof. Every finite index prime ideal P is maximal, therefore P^{n} is primary and hence P^{n} = P^{(n)} for all n; and the only height 1 prime ideals P in a Krull ring with [R : P^{(n)}] infinite for some n are those of infinite index. □

Finally, when A = R in the following statement, we retrieve Gilmer’s [1] result.

Corollary 2. If R is a Dedekind ring with finite residue fields, A a subring of R and S ⊆ A then F_{R}(S) = F_{R}(A) if and only if S contains a complete set of residues of P^{n} ∩ A in A for every prime ideal P of R and every n ∈ ℕ.

References

1. R. Gilmer, Sets That Determine Integer-Valued Polynomials, J. Number Theory 33 (1989), 95–100.

2. H. Matsumura, “Commutative ring theory,” Cambridge University Press, 1986.

3. D. L. McQuillan, On a Theorem of R. Gilmer, J. Number Theory 39 (1991), 245–250.

4. A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. reine angew. Mathematik, 149 (1919) 117–124.

5. G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. reine angew. Mathematik, 149 (1919) 97–116.

Institut für Mathematik (C) Technische Universität Graz Kopernikusgasse 24 A-8010 Graz, Austria

e-mail: frisch@blah.math.tu-graz.ac.at