INTEGER-VALUED POLYNOMIALS ON ALGEBRAS A SURVEY

SOPHIE FRISCH

Abstract. We compare several different concepts of integer-valued polynomials on algebras and collect the few results and many open questions to be found in the literature. (2000 Math. Subj. Classification: Primary 13F20; Secondary 16S50, 13B25, 13J10, 11C08, 11C20)

1. Introduction

Let D be a domain with quotient field K. The popular ring of integer-valued polynomials Int(D) = {f ∈ K[x] | f(D) ⊆ D} has been generalized to polynomials acting on non-commutative algebras in different ways by different authors. Some consider polynomials with coefficients in K that map a given D-algebra to itself. For instance, Loper [5] and the present author [2, 3] have investigated polynomials with rational coefficients mapping n × n integer matrices to integer matrices.

Others consider polynomials with coefficients in a non-commutative K-algebra that map a given D-subalgebra to itself. For instance, Werner [7] has investigated polynomials with coefficients in the rational quaternions mapping integer quaternions to integer quaternions; Werner [6] and the present author [2] have looked at polynomials with coefficients in M_{n}(K) mapping matrices in M_{n}(D) to matrices in M_{n}(D).

Before we give a precise definition of two types of rings of integer-valued polynomials on algebras, a few examples (in one variable). For lack of a better idea, we write the first kind of integer-valued polynomial rings, those with coefficients in K, with parentheses: Int_{D}(A), and the second kind, those with coefficients in a K-algebra, with square brackets: Int_{D}[A]. Throughout this paper, D is an integral domain, not a field, with quotient field K.

Example 1.1. For fixed n ∈ ℕ, consider

Int_{D}(M_{n}(D)) = {f ∈ K[x] | ∀C ∈ M_{n}(D) : f(C) ∈ M_{n}(D)}
Int_{D}[M_{n}(D)] = {f ∈ (M_{n}(K))[x] | ∀C ∈ M_{n}(D) : f(C) ∈ M_{n}(D)}.

Example 1.2. Let Q = ℚ + ℚi + ℚj + ℚk be the ℚ-algebra of rational quaternions and L the ℤ-subalgebra of Lipschitz quaternions ℤ + ℤi + ℤj + ℤk.

Int_{ℤ}(L) = {f ∈ ℚ[x] | ∀z ∈ L : f(z) ∈ L}
Int_{ℤ}[L] = {f ∈ Q[x] | ∀z ∈ L : f(z) ∈ L}

Example 1.3. Let G be a finite group, K(G) and D(G) group rings.

Int_{D}(D(G)) = {f ∈ K[x] | ∀z ∈ D(G) : f(z) ∈ D(G)}
Int_{D}[D(G)] = {f ∈ K(G)[x] | ∀z ∈ D(G) : f(z) ∈ D(G)}

Example 1.4. Let D ⊆ A be Dedekind rings with quotient fields K ⊆ F.

Int_{D}(A) = {f ∈ K[x] | f(A) ⊆ A}.

Convention 1.5. Let D be a domain and not a field, K the quotient field of D, and A a torsion-free D-algebra that is finitely generated as a D-module.

Since A is faithful, we have an isomorphic copy of D embedded in A (by d ↦ d1_{A}). Let B = K ⊗_{D} A (canonically isomorphic to the ring of fractions A_{D\{0}}). Then the natural homomorphisms ι_{K} : K → K ⊗_{D} A, k ↦ k ⊗ 1_{A} and ι_{A} : A → K ⊗_{D} A, a ↦ 1_{K} ⊗ a allow us to evaluate in B polynomials with coefficients in K or B at arguments in A, and we define:

Int_{D}(A) = {f ∈ K[x] | ∀a ∈ A : f(a) ∈ A}
Int_{D}[A] = {f ∈ (K ⊗_{D} A)[x] | ∀a ∈ A : f(a) ∈ A}

Note that ι_{K} and ι_{A} are injective whenever A is a torsion-free D-module. To exclude unwanted cases such as A = K we require K ∩ A = D (or, more precisely, ι_{K}(K) ∩ ι_{A}(A) = ι_{A}(D)).

Note that K ∩ A = D implies

Int_{D}(A) ⊆ Int(D) = {f ∈ K[x] | f(D) ⊆ D}.

With the conventions above, Int_{D}(A) is easily seen to be a ring. In particular, Int_{D}(A) is closed with respect to multiplication, because (fg)(a) = f(a)g(a) for all a ∈ A and f, g ∈ K[x]. By the same token, Int_{D}[A] is a ring for commutative A. The argument involving subsitution homomorphism works only in the commutative case, however. For non-commutative A, multiplicative closure of Int_{D}[A] is not evident. We will look into this in the next section.

2. Non-commutatve coefficient rings

Theorem 2.1 (Werner [6]). If A is finitely generated by units as a D-algebra, then Int_{D}[A] is closed under multiplication and hence a ring.

Proof. Let f(x) = ∑_{k} β_{k}x^{k} and g(x) be in Int_{D}[A] and α ∈ A. To show (fg)(α) ∈ A, we first check the special case where g = u, a unit in A:

(fu)(α) = ∑_{k} β_{k}uα^{k} = ∑_{k} β_{k}(uαu^{−1})^{k}u = f(uαu^{−1})u ∈ A.

Now for general f, g ∈ Int_{D}[A]:

(fg)(α) = ∑_{m,l} β_{m}γ_{l}α^{m+l} = ∑_{m} β_{m}(∑_{l} γ_{l}α^{l})α^{m} = ∑_{m} β_{m}g(α)α^{m}.

Expressing g(α) as a D-linear combination of units u_{1}, …, u_{n} of A,

g(α) = d_{1}u_{1} + … + d_{n}u_{n},

yields

(fg)(α) = ∑_{m} β_{m}(∑_{j=1}^{n} d_{j}u_{j})α^{m} = ∑_{j=1}^{n} d_{j} ∑_{m} β_{m}u_{j}α^{m} = ∑_{j=1}^{n} d_{j}(fu_{j})(α).

Since d_{j} ∈ D and each (fu_{j})(α) is in A, it follows that (fg)(α) is in A. □

Remark 2.2. In all three non-commutative examples in the introduction, A is generated as a D-module by units, and Int_{D}[A] is a therefore a ring. In example 1.1, for instance, the free D-module M_{n}(D) of dimension n^{2} has the following basis (suggested by L. Vaserstein) consisting of matrices of determinant 1: let E_{i,j}(λ) for i ≠ j denote the elementary matrix with ones on the diagonal, λ in position (i, j) and zeros elsewhere. As basis, take the n^{2} − n elementary matrices E_{i,j}(1) for i ≠ j, together with the n products of two elementary matrices E_{i,i+1}(1)E_{i+1,i}(−1) for 1 ≤ i ≤ n (with indices mod n, i.e., n + 1 = 1).

One of the rings of the form Int_{D}[A] for non-commutative A that have been examined in some detail is Int_{ℤ}[L], the ring of polynomials with coefficients in the rational quaternions mapping integer quaternions to integer quaternions. Werner [7] has shown Int_{D}[A] to be non-Noetherian, and has exhibited some prime ideals.

In his forthcoming paper [6], Werner explores Int_{D}[M_{n}(D)], and shows that every ideal of this ring is generated as a left M_{n}(D)-module by elements of K[x]. Using ideas from [6], one can show more, however: the ring Int_{D}[M_{n}(D)] of polynomials with coefficients in M_{n}(K) that map every matrix in M_{n}(D) to a matrix in M_{n}(D) is isomorphic to the ring of n × n matrices over the ring Int_{D}(M_{n}(D)) of polynomials in K[x] that map every matrix in M_{n}(D) to a matrix in M_{n}(D).

Theorem 2.3 ( [2]). Let

Int_{D}(M_{n}(D)) = {f ∈ K[x] | ∀C ∈ M_{n}(D) : f(C) ∈ M_{n}(D)},
Int_{D}[M_{n}(D)] = {f ∈ (M_{n}(K))[x] | ∀C ∈ M_{n}(D) : f(C) ∈ M_{n}(D)}.

We identify Int_{D}[M_{n}(D)] with its isomorphic image under the natural ring isomorphism

φ: (M_{n}(K))[x] → M_{n}(K[x]), ∑_{k}(a_{ij}^{(k)})_{1≤i,j≤n} x^{k} ↦ (∑_{k} a_{ij}^{(k)}x^{k})_{1≤i,j≤n} .

Then

Int_{D}[M_{n}(D)] = M_{n}(Int_{D}(M_{n}(D))).

Corollary 2.4. Under the identification of Int_{D}[M_{n}(D)] with its isomorphic image in M_{n}(K[x]), the ideals of Int_{D}[M_{n}(D)] are precisely the sets of the form M_{n}(I), where I is an ideal of Int_{D}(M_{n}(D)). Prime ideals of Int_{D}[M_{n}(D)] correspond to prime ideals of Int_{D}(M_{n}(D)) and vice versa.

Our definition of prime ideal for a possibly non-commutative ring is: a two-sided ideal P ≠ R, such that for any two-sided ideals A, B of R, AB ⊆ R implies A ⊆ P or B ⊆ P.

It might be interesting to generalize Theorem 2.3 to other rings of integer-valued polynomials on a D-algebra A with coefficients in a non-commutative K-algebra B. Given a matrix representation B ⊆ M_{n}(K), we can identify the ring Int_{D}[A] ⊆ B[x] of polynomials with coefficents in B, integer-valued on A, with its image in M_{n}(K[x]) under the isomorphism of (M_{n}(K))[x] with M_{n}(K[x]).

• Starting with a matrix representation B ⊆ M_{n}(K), is the isomorphic image of Int_{D}[A] ⊆ (M_{n}(K))[x] embedded in M_{n}(K[x]) a matrix algebra over a ring of integer-valued polynomials with coefficents in K?

3. The Spectrum

We now return to commuting coefficents and describe the spectrum of Int_{D}(A). If A is a commutative D-algebra, we also consider polynomials is several variables and define

Int_{D}^{n} = {f ∈ K[x_{1}, …, x_{n}] | ∀a ∈ A^{n} : f(a) ∈ A}.

Prime ideals lying over a prime P of infinite index of D are easy to describe: they all come from prime ideals of D_{P} [x] (or D_{P} [x_{1}, …, x_{n}], for Int_{D}^{n}(A)), since Int_{D}(A) ⊆ Int(D) ⊆ D_{P} [x] (and Int_{D}^{n}(A) ⊆ Int(D^{n}) ⊆ D_{P} [x_{1}, …, x_{n}]) whenever [D : P] = ∞ (cf. [1]).

Concerning primes lying over a maximal ideal M of finite index of D, they have been characterized for one-dimensional Noetherian D in [2]. For commutative A, they look just like the maximal ideals of Int(D).

Note that the somewhat technical condition MA_{M} ∩ A = MA is satisfied in two natural cases, firstly, if A is a free D-module, and secondly, if D ⊆ A is an extension of Dedekind rings.

Theorem 3.1 ( [2]). Let D be a domain, A a commutative torsion-free D-algebra finitely generated as a D-module, M a finitely generated maximal ideal of D of finite index and height one, such that MA_{M} ∩ A = MA, and n ∈ ℕ.

Then every prime ideal of Int_{D}^{n}(A) lying over M is maximal, and of the form

P_{a} = {f ∈ Int_{D}^{n}(A) | f(a) ∈ P },

for some a ∈ Â (the M-adic completion of A) and P a maximal ideal of Â with P ∩ D = M.

In the case of a non-commutative D-algebra A, the images of elements a ∈ Â under Int_{D}(A) play a rôle in the description of the maximal ideals lying above M. If the exact image Int_{D}(A)(a) is not known, it can be replaced by a commutative ring R_{a} between Int_{D}(A)(a) and Â.

Theorem 3.2 ( [2]). Let D be a domain, A a torsion-free D-algebra finitely generated as a D-module, M a finitely generated maximal ideal of D of finite index and height one, such that MA_{M} ∩ A = MA.

The prime ideals of Int_{D}(A) lying over M are precisely the ideals of the form

P_{a} = {f ∈ Int_{D}(A) | f(a) ∈ P },

where a ∈ Â (the M-adic completion of A), and P is a maximal ideal of Int_{D}(A)(a) (the image of a under Int_{D}^{n}(A)) with P ∩ D = M.

We can replace Int_{D}(A)(a) by a commutative ring R_{a} with Int_{D}(A)(a) ⊆ R_{a} ⊆ Â for the simple reason that every extension of finite commutative rings, in particular the ring extension Int_{D}(A)(a)/(Int_{D}(A)(a) ∩ MÂ) ⊆ R_{a}/(R_{a} ∩ MÂ) satisfies “lying over”.

Corollary 3.3. Under the hypotheses of of Theorem 3.2, suppose we are given, for every a ∈ Â, a commutative ring R_{a} with Int_{D}(A)(a) ⊆ R_{a} ⊆ Â.

Then the prime ideals of Int_{D}(A) are precisely the ideals of the form

P_{a} = {f ∈ Int_{D}(A) | f(a) ∈ P },

where a ∈ Â and P is a maximal ideal of R_{a} lying over M.

For A = M_{n}(D), and a ∈ A, the image of a under Int(A)(a) is just D[a], and for a general a ∈ Â, the image of a under Int(A)(a) is contained in D̂[a] (cf. [2]), so that we may take R_{a} = D̂[a] in Corollary 3.3. For other algebras, the question is open:

• is there a simple description of the image of an element a ∈ Â under Int_{D}(A)? Another property of the ring of integer-valued polynomials on matrices is waiting

for generalization. If D is a domain with zero Jabobson radical, such as, for instance, a Dedekind ring with infinitely many maximal ideals, then the subset C of M_{n}(D) consisting of the companion matrices of all monic irreducible polynomials in D is a polynomially dense subset of M_{n}(D), i.e., every polynomial f ∈ K[x] with f(C) ∈ M_{n}(D) for every C ∈ C is in Int_{D}(M_{n}(D)). This prompts the question, for a general D-algebra A,

• does A have a polynomially dense subset of elements with irreducible minimal polynomial in K[x]?

4. A non-triviality criterion

For rings of integer valued polynomials with coefficients in a field, of the type

Int_{D}(A) = {f ∈ K[x] | f(A) ⊆ A},

or, for commutative A,

Int_{D}^{n}(A) = {f ∈ K[x_{1}, …, x_{n}] | ∀a_{1}, …, a_{n} ∈ A : f(a_{1}, …, a_{n}) ∈ A},

we have the inclusions

D[x] ⊆ Int_{D}(A) ⊆ Int(D) ⊆ K[x],

and similarly for several variables. As before, D is a domain with quotient field K, A a torsion-free D-algebra finitely generated as a D-module, and evaluation of polynomials is performed in B = K ⊗_{D} A. As noted in the introduction, we also require (of the homomorphic images in B) that K ∩ A = D.

Int_{D}(A) is considered trivial if Int_{D}(A) = D[x]. We will see that the non-triviality criterion for Int(D) = {f ∈ K[x] | f(D) ⊆ D} for Noetherian D ( [1] Thm. I.3.14) carries over to Int_{D}(A).

Lemma 4.1. Let A be a torsion-free D-algebra that is finitely generated as a Dmodule, and let n ∈ ℕ. If there exists a proper ideal of D of the form I = (b :_{D} c) (with b, c ∈ D) of finite index, then Int_{D}^{n}(A) ≠ D[x_{1}, …, x_{n}].

Proof. Say A is generated by d elements as a D-module. Then every element of A is integral of degree at most d over D. Given I = (b :_{D} c) ≠ D of finite index, let f ∈ D[x] be a monic polynomial that is divisible modulo I[x] by every monic polynomial of degree at most d. Then for every a ∈ A, f(a) ∈ IA, and hence \frac{c}{b}f(a) ∈ A. If follows that \frac{c}{b}f(x) is in Int_{D}(A) (as well as in Int_{D}^{n}(A) for all n ≥ 1), but not in D[x], since its leading coefficient \frac{c}{b} is not in D. □

Lemma 4.2. If, for some n ∈ ℕ, Int_{D}^{n}(A) ≠ D[x_{1}, …, x_{n}] then there exists a proper ideal of D of the form I = (b :_{D} c) (with b, c ∈ D) such that every prime ideal P of D containing I is of finite index.

Proof. Let b, c ∈ D such that k = \frac{c}{b} ∉ D occurs as a coefficient of a polynomial in Int_{D}^{n}(A). If P is a prime ideal of infinite index in D, then Int_{D}^{n}(A) ⊆ D_{P} [x_{1}, …, x_{n}]; so there exists some s ∈ D \ P with sk ∈ D, i.e., with s ∈ (b :_{D} c). This means that (b :_{D} c) is not contained in any prime ideal of infinite index. □

It is easy to see that, for arbitrary fixed b ∈ D, an ideal that is maximal among proper ideals of the form (b : d) (with d ∈ D) is prime. In a Noetherian domain D therefore, every proper ideal I = (b : c) is contained in a prime ideal P = (b : d). This shows that for a Noetherian domain D and a D-algebra A whose elements are integral of bounded degree over D, the necessary and the sufficient condition for Int_{D}(A) ≠ D[x] (in 4.1 and 4.2, respectively) are each equivalent to: D has a prime ideal of finite index of the form P = (b : d).

If, given an ideal I of D, we call a prime ideal of the form (I :_{D} d) (with d ∈ D) an associated prime ideal of I then our criterion for non-triviality of Int_{D}^{n}(A) in the Noetherian case becomes:

Theorem 4.3. Let D be a Noetherian domain and A a torsion-free D-algebra that is finitely generated as a D-module and let n ∈ ℕ. Then Int_{D}^{n}(A) ≠ D[x_{1}, …, x_{n}] if and only if D has a prime ideal of finite index that is an associated prime of a principal ideal of D.

A different question of non-triviality is, whether Int_{D}(A) is properly contained in Int(D). (Recall that Int_{D}(A) ⊆ Int(D) follows from our convention K ∩ A = D.) Let K be a number field and O_{K} its ring of algebraic integers. It has been shown by Halter-Koch and Narkiewicz [4] that Int_{ℤ}(O_{K}) is always properly contained in Int(ℤ). For general D and A it is an open question,

• under what hypotheses is Int_{D}(A) ⊊ Int(D)?

5. Prüfer or not Prüfer

For rings of integer-valued polynomials on algebras of the type

Int_{ℤ}(A) = {f ∈ ℚ[x] | f(A) ⊆ A},

for a ℤ-algebra A, the big question is, what are criteria for Int_{ℤ}(A) to be Prüfer, or just to be integrally closed?

In some interesting special cases Loper [5] has the answer:

Theorem 5.1 (Loper [5]).

(1) Let O_{K} be the ring of algebraic integers in the number field K. Then Int_{ℤ}(O_{K}) is Prüfer.

(2) Let M_{2}(ℤ) be the ring of 2 × 2 integer matrices, then Int_{ℤ}(M_{2}(ℤ)) is not Prüfer.

(3) Let L be the ring of integer (Lipschitz) quaternions. Then Int_{ℤ}(L) is not Prüfer.

In cases 2 and 3, Loper shows that the ring in question is not Prüfer by exhibiting an overring that is not integrally closed. For any non-commutative ℤ-algebra A, such as A = M_{n}(ℤ) or A = L, this prompts the following questions:

• Is Int_{ℤ}(A) integrally closed?

• What is its integral closure?

• Is the integral closure Prüfer?

References

[1] P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, vol. 48 of Mathematical Surveys and Monographs, Amer. Math. Soc., 1997.

[2] S. Frisch, Integer-valued polynomials on algebras, preprint.

[3] S. Frisch, Polynomial separation of points in algebras, in Arithmetical Properties of Commutative Rings and Monoids (Chapel Hill Conf. 2003), S. Chapman, ed., Dekker, 2005, 249–254.

[4] F. Halter-Koch and W. Narkiewicz, Commutative rings and binomial coefficients, Mh. Math 114 (1992), 107–110.

[5] K. A. Loper, A generalization of integer-valued polynomial rings, preprint.

[6] N. J. Werner, Integer-valued polynomials over matrix rings, preprint.

[7] N. J. Werner, Integer-valued polynomials over quaternion rings, J. Algebra (2010), 1754–1769.

Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria

  E-mail address: frisch@TUGraz.at