Appeared in J. Algebra 373 (2013) 414-425

INTEGER-VALUED POLYNOMIALS ON ALGEBRAS

Sophie Frisch

                          frisch@TUGraz.at

Abstract. Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine spectrum and Krull dimension of the ring Int_{D}(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in M_{n}(K), the K-algebra of n × n matrices, that map every matrix in M_{n}(D) to a matrix in M_{n}(D).

2000 MSC: Primary 13F20; Secondary 16S50, 13B25, 13J10, 11C08, 11C20.

1. Introduction

Let D be a domain with quotient field K and A a D-algebra, such as, for instance, a group ring D(G) or the matrix algebra M_{n}(D).

We are interested in the rings of polynomials

Int_{D}(A) = {f ∈ K[x] | f(A) ⊆ A},

and, if A is commutative,

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

Elements of the D-algebra A are plugged into polynomials with coefficients in K via the canonical homomorphism ι_{A} : A → K ⊗_{D} A, ι_{A}(a) = 1 ⊗ a.

INTEGER-VALUED POLYNOMIALS ON ALGEBRAS

In the special case A = D these rings are known as rings of integer-valued polynomials, cf. [3]. They provide natural examples of non-Noetherian Prüfer rings [5, 11], and have been used for proving results on the n-generator property in Prüfer rings [2]. Also, integer-valued polynomials are useful for polynomial interpolation of functions from D to D [8, 4], and satisfy other interesting algebraic conditions such as analogues of Hilbert’s Nullstellensatz [3, 9].

These desirable properties of rings of integer-valued polynomials have motivated the generalization to polynomials with coefficients in K acting on a Dalgebra A [10, 12]. So far, not much is known about rings of integer-valued polynomials on algebras. We know that they behave somewhat like the classical rings of integer-valued polynomials if the D-algebra A is commutative. For instance, Loper and Werner [12] have shown that Int_{ℤ}(O_{K}) is Prüfer. If A is non-commutative, however, the situation is radically different. For instance, Int_{ℤ}(M_{2}(ℤ)) is not Prüfer [12], and is far from allowing interpolation [10].

We will describe the spectrum of Int_{D}(A), for a one-dimensional Noetherian ring D and a finitely generated torsion-free D-algebra A, in the hope that this will facilitate further research. We will investigate more closely the special case of A = M_{n}(D): we determine a polynomially dense subset of M_{n}(D) and describe the image of a given matrix under the ring Int_{D}(M_{n}(D)).

A different ring of integer-valued polynomials on the matrix algebra M_{n}(D), consisting of polynomials with coefficients in M_{n}(K) that map matrices in M_{n}(D) to matrices in M_{n}(D), has been introduced by Werner [13]. We will show that it is isomorphic to the algebra of n × n matrices over “our” ring Int_{D}(M_{n}(D)) of integer-valued polynomials on M_{n}(D) with coefficients in K.

Before we give a precise definition of the kind of D-algebra A for which we will investigate Int_{D}(A), a few examples. D is always a domain with quotient field K, and not a field.

1.1 Example. For fixed n ∈ ℕ, let A = M_{n}(D) be the D-algebra of n×n matrices with entries in D and

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

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

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

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

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

If G is commutative, we also consider

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

for n ∈ ℕ, where D(G)^{n} = D(G) × … × D(G) denotes the Cartesian product of n copies of D(G).

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

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

1.5 Notation and conventions. Throughout this paper, D is a domain and not a field, K the quotient field of D, and A a torsion-free D-algebra finitely generated as a D-module. Since A is faithful, there is an isomorphic copy of D embedded in A by d ↦ d1_{A}, and we may assume D ⊆ A.

Now let B = K ⊗_{D} A. The natural homomorphisms ι_{K} : K → K ⊗_{D} A, ι_{K}(k) = k ⊗ 1 and ι_{A} : A → K ⊗_{D} A, ι_{A}(a) = 1 ⊗ a, are injective, since A is a torsion-free D module. We identify K and A with their isomorphic copies in B, which allows us to evaluate polynomials with coefficients in K at arguments in A, and define

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

and for n ∈ ℕ_{0}

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

To exclude pathological cases we require K ∩ A = D.

1.6 Remark. Instead of B = K ⊗_{D} A, we could look at the canonically isomorphic A_{K\{0}}, the ring of fractions of A with denominators in K \ {0}. The natural homomorphisms ι_{K} : K → A_{K\{0}} and ι_{A} : A → A_{K\{0}} then take the form ι_{K}(\frac{c}{d}) = \frac{c1_{A}}{d} and ι_{A}(a) = \frac{a}{1} .

1.7 Convention regarding polynomials in several variables. For non-commutative A and n > 1, Int_{D}^{n}(A) is a priori not closed under multiplication and therefore in general not a ring. With the exception of the following section on I-adic continuity, we will only consider polynomial functions in several variables if the D-algebra A is commutative.

From section 3 onward, statements about Int_{D}^{n}(A) with unspecified n and A are meant as follows: if A is commutative, let n ∈ ℕ_{0}, if A is non-commutative, assume n ≤ 1.

1.8 Remark. Note that K ∩ A = D implies

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

and for commutative A, also

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

and Int_{D}^{0}(A) = K ∩ A = D.

2. Continuity

I-adic continuity of the function f: D^{n} → D arising from a classical integervalued polynomial f ∈ Int(D^{n}) = {f ∈ K[x_{1}, …, x_{n}] | f(D^{n}) ⊆ D} has been shown, for an arbitrary ideal I of an arbitrary domain D, in Prop. 1.4 of [4]. (The proof there is for one variable, but clearly generalizable to several variables.)

To establish I-adic continuity of integer-valued polynomials on algebras, we will briefly look at polynomials in several non-commuting variables. If our algebra A is non-commutative, this becomes necessary, even if we are only interested in integer-valued polynomials in one variable: if we consider f(x + y) − f(y) as a polynomial in two variables, and we still want substitution of elements from A for x and y to be a homomorphism, we must turn to non-commuting variables.

2.1 Definition. Let D be a domain with quotient-field K. Let K⟨x_{1}, …, x_{n}⟩ be the free associative K-algebra generated by x_{1}, …, x_{n} (in other words, the semigroup-ring K(S), where S is the free semigroup generated by x_{1}, …, x_{n}). If A is a torsion-free D-algebra, we evaluate polynomials in K⟨x_{1}, …, x_{n}⟩ at arguments in B = K ⊗_{D} A and thus associate a polynomial function f: B^{n} → B to every f ∈ K⟨x_{1}, …, x_{n}⟩. Such polynomials as map arguments in A^{n} to values in A we call integer-valued on A.

From the theory of PID-rings (polynomial identity rings), it is easy to garner non-trivial examples of integer-valued polynomials in several non-commuting variables. For instance, if p is prime and n ≥ 1, then a polynomial in ℚ⟨x_{1}, …, x_{np}⟩, but not in ℤ⟨x_{1}, …, x_{np}⟩, that takes every np-tuple of n × n integer matrices to an integer matrix is

f(x_{1}, …, x_{np}) = \frac{1}{p} ∑_{π∈S_{np}} x_{π(1)}x_{π(2)} … x_{π(np)}.

(This follows from [7] or [14].) In this paper, we will not consider polynomials in non-commuting variables except in the following theorem and its corollaries.

2.2 Theorem. Let D be a domain with quotient-field K and A a torsion-free D-algebra. For every f ∈ K⟨x_{1}, …, x_{n}⟩ integer-valued on A, the polynomial function f: A^{n} → A is uniformly I-adically continuous for every ideal I of D.

Proof. Fix i and let d be the degree of f in x_{i}. We will show that for every b ∈ I^{d}A and every (a_{1}, …, a_{n}) ∈ A^{n},

(∗)  f(a_{1}, …, a_{i} + b, …, a_{n}) − f(a_{1}, …, a_{i}, …, a_{n}) ∈ IA.

For d = 0, or if f is the zero-polynomial, this is obvious. The polynomial f is uniquely representable as f = f_{1} + f_{2}, where x_{i} doesn’t occur in f_{1}, x_{i} occurs in every monomial in the support of f_{2}, and f_{2} has the same degree in x_{i} as f. Since f_{1} = f(x_{1}, …, x_{i−1}, 0, x_{i+1}, …, x_{n}) and f_{2} = f − f_{1}, both f_{1} and f_{2} are integer-valued. We can show (∗) separately for f_{1} and f_{2}. As (∗) holds for f_{1} and arbitrary b ∈ A, we have reduced to the case f = f_{2}, i.e., when x_{i} occurs in every monomial in the support of f.

Also, it suffices to show (∗) for b = t_{d}t_{d−1} … t_{1}c with t_{k} ∈ I and c ∈ A, because every element of I^{d}A is a finite sum of elements of this form.

By considering

g(x_{1}, …, x_{n}, z) = f(x_{1}, …, x_{i} + z, …, x_{n}) − f(x_{1}, …, x_{i}, …, x_{n})

we can reduce our task to showing for every g(x_{1}, …, x_{n}, z) ∈ K⟨x_{1}, …, x_{n}, z⟩ of degree d ≥ 1 in z, which maps A^{n+1} to A, satisfies g(x_{1}, …, x_{n}, 0) = 0, and such that z occurs in every monomial in the support of g:

(∗∗)  g(a_{1}, …, a_{n}, t_{d} … t_{1}c) ∈ IA whenever t_{1}, …, t_{d} ∈ I and c ∈ A.

We show (∗∗) by induction on d. Let d ≥ 1. Consider

h(x_{1}, …, x_{n}, y) = g(x_{1}, …, x_{n}, t_{d}y) − t_{d}^{d} g(x_{1}, …, x_{n}, y).

h satisfies h(x_{1}, …, x_{n}, 0) = 0, is of degree at most d − 1 in y, and y occurs in every monomial in the support of h. Now h(a_{1}, …, a_{n}, t_{d−1} … t_{1}c) ∈ IA for all t_{1}, …, t_{d−1} ∈ I and c ∈ A, either by induction hypothesis or because h is the zeropolynomial. Therefore g(a_{1}, …, a_{n}, t_{d}t_{d−1} … t_{1}c) = h(a_{1}, …, a_{n}, t_{d−1} … t_{1}c) + t_{d}^{d} g(a_{1}, …, a_{n}, t_{d−1} … t_{1}c) is in IA, for all t_{1}, …, t_{d} ∈ I and c ∈ A. □

Returning to commuting variables, we conclude:

2.3 Corollary. For any ideal I of D, and every f ∈Int_{D}^{n}(A) the function f: A^{n} →A is uniformly I-adically continuous.

2.4 Corollary. If M is a maximal ideal of D, Â the M -adic completion of A, and f ∈ Int_{D}^{n}(A), then the function f: A^{n} → A defined by f extends uniquely to a M -adically continuous function f : Â^{n} → Â.

3. A few technicalities

This section contains lemmata needed for the investigation of the spectrum of Int_{D}(A) and, for commutative A, of Int_{D}^{n}(A). From now on, all statements about Int_{D}^{n}(A) are subject to the convention: if A is non-commutative, assume n ≤ 1.

3.1 Lemma. Let D be a domain and P a finitely generated prime ideal of height 1. For every non-zero p ∈ P there exist m ∈ ℕ and s ∈ D \ P such that sP^{m} ⊆ pD.

Proof. Let p ∈ P. In the localization D_{P}, P_{P} is the radical of (q) for every non-zero q ∈ P_{P}. Therefore, since P (and hence P_{P}) is finitely generated, there exists m ∈ ℕ with P_{P}^{m} ⊆ pD_{P} and in particular P^{m} ⊆ pD_{P}.

The ideal P^{m} is also finitely generated, by p_{1}, …, p_{k}, say. Let a_{i} ∈ D_{P} with p_{i} = pa_{i}. By considering the fractions a_{i} = r_{i}/s_{i} (with r_{i} ∈ D and s_{i} ∈ D \ P), and setting s = s_{1} · . . . · s_{k}, we see that sP^{m} ⊆ pD as desired. □

3.2 Definition. If S ⊆ B^{n} and T ⊆ B, let

Int_{D}^{n}(S, T) = {f ∈ K[x_{1}, …, x_{n}] | f(S) ⊆ T }.

3.3 Lemma. Let D be a domain and P a finitely generated prime ideal of height 1. Then every prime ideal Q of Int_{D}^{n}(A) with Q ∩ D = P contains Int_{D}^{n}(A, P A).

Proof. Let f ∈ Int_{D}^{n}(A, P A). By dint of Lemma 3.1, there are m ∈ ℕ, p ∈ P and s ∈ D \ P such that sP^{m} ⊆ pD. Then sf^{m} ∈ Int_{D}^{n}(A, pA) = p Int_{D}^{n}(A) ⊆ Q. As Q is prime and s ∉ Q, we conclude that f ∈ Q. □

3.4 Lemma. Let A be a D-algebra that is finitely generated as a D-module, M a maximal ideal of finite index in D and Â the M -adic completion of A. For all a ∈ Â^{n}, the ideal (MÂ)_{a} = {f ∈ Int_{D}^{n}(A) | f(a) ∈ MÂ} is of finite index in Int_{D}^{n}(A).

Proof. Since A is a finitely generated D-module, Â is a finitely generated D̂- module. As M̂ is of finite index in D̂, Â/MÂ is finite. Let Int_{D}^{n}(A)(a) denote the image of a under Int_{D}^{n}(A). Then Int_{D}^{n}(A)(a)/(MÂ ∩ Int_{D}^{n}(A)(a)), as a subring of Â/MÂ, is finite.

Let b_{1}, …, b_{m} be a system of representatives of Int_{D}^{n}(A)(a) modulo MÂ ∩ Int_{D}^{n}(A)(a), and for 1 ≤ i ≤ m let f_{i} ∈ Int_{D}^{n}(A) with f_{i}(a) = b_{i}. Then for every f ∈ Int_{D}^{n}(A), exactly one of the differences f − f_{i} is in (MÂ)_{a}, which means that {f_{1}, …, f_{m}} is a complete system of residues of Int_{D}^{n}(A)(a) modulo (MÂ)_{a}. We have shown [Int_{D}^{n}(A) : (MÂ)_{a}] = [Int_{D}^{n}(A)(a) : MÂ ∩ Int_{D}^{n}(A)(a)] ≤ [Â: MÂ] □

4. Primes lying over a height one maximal ideal of finite index.

In this section we determine the prime ideals of Int_{D}(A) (and, for commutative A, of Int_{D}^{n}(A)) lying over a height one prime ideal of finite index of D. Prime ideals lying over a prime of infinite index in D will be characterized in the next section.

4.1 General hypotheses in this section.

(i) D is a domain and A a torsion-free D-algebra, finitely generated as a Dmodule;

(ii) M is a finitely generated maximal ideal of height 1 and finite index in D; (iii) MA_{M} ∩ A = MA. (Note that MA_{M} ∩ A = MA is satisfied in two important

cases: if A is a free D-module, and if D ⊆ A is an extension of Dedekind rings.)

We denote the M -adic completions of D and A by D̂ and Â. Since M is finitely generated, M -adic and M̂ -adic topologies coincide on D̂ and on Â.

4.2 Lemma. The hypotheses of 4.1 imply (iv) MÂ ∩ A = MA.

(v) D̂ and Â are compact.

Proof. (iv) Whenever M is a finitely generated maximal ideal of height 1 in a domain D and A a finitely generated D-module, the equality MÂ∩A = MA_{M} ∩A holds, by [1, Chapter III, §2.12, Prop. 16] combined with [1, §3.5, Cor. 1 of Prop. 9].

(v) The ring D̂ is compact because M is of finite index and finitely generated, which implies that all powers of M are of finite index. Â then is compact because it is finitely generated as a D̂-module. □

4.3 Notation. Recall our convention that we only allow n > 1 in Int_{D}^{n}(A) if A is commutative; for non-commutative A, n = 1 is assumed.

The image of a ∈ Â under Int_{D}^{n}(A) and Int_{D}(A) we denote as follows:

Int_{D}^{n}(A)(a) = {f(a) | f ∈ Int_{D}^{n}(A)} and Int_{D}(A)(a) = {f(a) | f ∈ Int_{D}(A)}.

If M is a maximal ideal of D and a ∈ Â^{n} let

(MÂ)_{a} = {f ∈ Int_{D}^{n}(A) | f(a) ∈ MÂ}.

If P is an ideal of a commutative ring between Int_{D}^{n}(a) and Â, let

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

4.4 Lemma. Under the hypotheses of 4.1, let Q be a prime ideal of Int_{D}^{n}(A) lying over M. Then there exists a ∈ Â such that

(MÂ)_{a} ⊆ Q.

In particular, Q is of finite index.

Proof. Suppose Q does not contain any (MÂ)_{a}. Then for every a∈Â^{n} there exists f ∈ (MÂ)_{a} \ Q. By Theorem 2.3, f is M -adically continuous, so there exists an M -adic neighborhood U of a, such that f(U) ⊆ MÂ. By compactness of Â^{n}, there exist finitely many a_{i} such that the corresponding neighborhoods U_{i} cover Â^{n}. Let g be the product of the polynomials f_{i} ∈ (MÂ)_{a_{i}} \ Q with f_{i}(U_{i}) ⊆ MÂ. For all a ∈ A, g(a) ∈ MÂ ∩ A = MA_{M} ∩ A = MA (by Lemma 4.2). Since Q is prime, g ∉ Q, and yet g ∈ Int_{D}^{n}(A, MA), a contradiction to Lemma 3.3. We have shown that Q contains some (MÂ)_{a}, which is of finite index by Lemma 3.4. □

In the special case A=D, the preceding lemma already concludes the characterization of primes of Int_{D}^{n}(D) lying above a maximal ideal M of finite index in D (a result of Chabert [6]), because then (MÂ)_{a} is (MD̂)_{a} = {f ∈ Int_{D}^{n}(D) | f(a) ∈ M̂}, a prime ideal of finite index, and hence maximal. Chabert, however, showed the other inclusion, Q ⊆ (MÂ)_{a}, after first showing independently that Q must be maximal [3, Prop. V.2.2].

4.5 Lemma. Under the hypotheses of 4.1, let Q be a prime ideal of Int_{D}^{n}(A) lying over M. For every a ∈ Â such that (MÂ)_{a} ⊆ Q, there exists a maximal ideal P of Int_{D}^{n}(A)(a) such that Q = P_{a}.

Proof. Since Â/MÂ is a finite ring, Int_{D}^{n}(A)(a)/(Int_{D}^{n}(A)(a) ∩ MÂ) is a finite commutative ring. Let P_{1}, …, P_{k} be the maximal ideals of Int_{D}^{n}(A)(a) containing Int_{D}^{n}(A)(a) ∩ MÂ.

Suppose Q is not contained in any (P_{i})_{a}, for 1 ≤ i ≤ k. Then, by prime avoidance, Q(a) = {f(a) | f ∈ Q} is not contained in ⋃_{i=1}^{k} P_{i}. Let f ∈ Q such that f(a) is not in any P_{i}. Then the residue class of f(a) is a unit in Int_{D}^{n}(A)(a)/(Int_{D}^{n}(A)(a) ∩ MÂ).

Replacing f by a suitable power of f (using the fact that the group of units of Int_{D}^{n}(A)(a)/(Int_{D}^{n}(A)(a) ∩ MÂ) is finite) we see that there exists f ∈ Q with f(a) ≡ 1 mod MÂ. It follows that 1 − f ∈ (MÂ)_{a} ⊆ Q and therefore 1 ∈ Q, a contradiction. □

4.6 Theorem. 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}^{n}(A) lying over M are precisely the ideals of the form

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

where a∈Â (the M -adic completion of A), and P is a maximal ideal of Int_{D}^{n}(A)(a) (the image of a under Int_{D}^{n}(A)) with P ∩ D = M. In particular, all primes of Int_{D}^{n}(A) lying over M are of finite index.

Proof. There exist primes of Int_{D}^{n}(A) lying over M, because (MÂ)_{a} with a ∈ Â is a proper ideal of Int_{D}^{n}(A) containing M.

If Q is a prime ideal of Int_{D}^{n}(A) lying over M, then Lemma 4.4 shows that there exists an element a ∈ Â such that (MÂ)_{a} is contained in Q, and that Q is of finite index. It then follows from Lemma 4.5 that Q = P_{a} for some maximal ideal P of Int_{D}^{n}(A)(a) satisfying M ⊆ Int_{D}^{n}(A)(a) ∩ MÂ ⊆ P, and hence P ∩ D = M.

Conversely, if P is a maximal ideal of Int_{D}^{n}(A)(a), then Int_{D}^{n}(A)/P_{a} is isomorphic to Int_{D}^{n}(A)(a)/P. Therefore P_{a} is a maximal ideal of Int_{D}^{n}(A)(a). □

It may happen that we do not know the exact image of a ∈ Â under Int_{D}^{n}(A), but do know a commutative ring R_{a} between Int_{D}^{n}(A)(a) and Â. In this case we should remember that R_{a}/(R_{a} ∩ MÂ) is a subring of the finite ring (Â/MÂ), and that therefore Int_{D}^{n}(A)(a)/(Int_{D}^{n}(A)(a) ∩ MÂ) ⊆ (R_{a}/R_{a} ∩ MÂ) is an extension of finite commutative rings. Since extensions of finite commutative rings satisfy “lying over”, every prime ideal of Int_{D}^{n}(A)(a) comes from a prime ideal of R_{a}, and we conclude:

4.7 Corollary. Under the hypotheses of of Theorem 4.6, suppose we have, for every a ∈ Â, a commutative ring R_{a} with Int_{D}^{n}(A)(a) ⊆ R_{a} ⊆ Â.

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

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

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

If A is a commutative D-algebra, we can take R_{a} = Â in Corollary 4.7 for all a ∈ Â, and we get the following simpler characterization of the primes of Int_{D}^{n}(A) lying over M:

4.8 Theorem. 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.

Then every prime ideal of Int_{D}^{n}(A) lying over M is 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 lying over M. In particular, every prime of Int_{D}^{n}(A) lying over M is of finite index.

5. Primes lying over prime ideals of infinite index

5.1 Lemma. Let A be a torsion-free D-algebra with K ∩ A = D. Let P be a prime ideal of infinite index in D and n ∈ ℕ. Then Int_{D}^{n}(A) ⊆ D_{P} [x_{1}, …, x_{n}].

Proof. More generally, we show that for any prime ideal P, a polynomial f ∈ Int_{D}^{n}(A) of degree less than [D:P] in every individual variable is in D_{P} [x_{1}, …, x_{n}]. We use induction on n. The case n = 0 is trivial: Int_{D}^{0}(D) = K ∩ A = D ⊆ D_{P}.

For n > 0 consider f as a polynomial in x_{n} with coefficients in K[x_{1}, …, x_{n−1}]. Let s ≤ [D : P] such that f is of degree strictly less than s in each x_{j}.

Choose d_{1}, …, d_{s} ∈ D ⊆ A pairwise incongruent mod P. For every i, the value of f at d_{i} (substituted for x_{n}) is a polynomial in Int_{D}^{n−1}(A) of degree less than s in each variable. Therefore f(x_{1}, …, x_{n−1}, d_{i}) ∈ D_{P} [x_{1}, …, x_{n−1}], by induction hypothesis.

Let g ∈ K(x_{1}, …, x_{n−1})[x_{n}] be the Lagrange interpolation polynomial with g(d_{i}) = f(x_{1}, …, x_{n−1}, d_{i}) (1 ≤ i ≤ s); then g ∈ D_{P} [x_{1}, …, x_{n−1}][x_{n}]. Since a polynomial (with coefficients in a domain) of degree less than s is determined by its values at s different arguments, we must have f = g ∈ D_{P} [x_{1}, …, x_{n−1}][x_{n}]. □

5.2 Corollary. Let A be a torsion-free D-algebra with K ∩ A = D. If all maximal ideals of D are of infinite index, then Int_{D}^{n}(A) = D[x_{1}, …, x_{n}].

Alternatively, we could have deduced the previous lemma from the corresponding fact for the ring of integer-valued polynomials over D [3, Prop. I.3.4, XI.1.10], since after all Int_{D}^{n}(A) ⊆ Int(D^{n}) = {f ∈ K[x_{1}, …, x_{n}] | f(D^{n}) ⊆ D}.

5.3 Lemma. Let A be a torsion-free D-algebra with K ∩ A = D. Let P be a prime ideal of infinite index in D and n ∈ ℕ. Then the prime ideals of Int_{D}^{n}(A) lying over P are precisely those of the form Q ∩ Int_{D}^{n}(A), where Q is a prime ideal of D_{P} [x_{1}, …, x_{n}] containing P D_{P} [x_{1}, …, x_{n}].

Proof. As D[x_{1}, …, x_{n}] ⊆ Int_{D}^{n}(A) ⊆ D_{P} [x_{1}, …, x_{n}] = D[x_{1}, …, x_{n}]_{(D\P)}, we have

D_{P} [x_{1}, …, x_{n}] = Int_{D}^{n}(A)_{(D\P)}

and therefore a bijective correspondence (given by lying over) exists between prime ideals of Int_{D}^{n}(A) whose intersection with D is contained in P and prime ideals of D_{P} [x_{1}, …, x_{n}]. □

5.4 Theorem. Let D be a Noetherian one-dimensional domain with finite residue fields and A a torsion-free D algebra, finitely generated as a D-module, such that for every maximal ideal M, MA_{M} ∩ D = M. If A is commutative, let n ∈ ℕ, for non-commutative A restrict to n = 1. Then Int_{D}^{n}(A) is (n + 1)-dimensional.

Proof. By Lemma 5.3, the prime ideals of Int_{D}^{n}(A) lying over (0) all come from prime ideals of K[x_{1}, …, x_{n}]. The primes of Int_{D}^{n}(A) lying over a maximal ideal M are all maximal and hence mutually incomparable. So dim(Int_{D}^{n}(A)) ≤ n + 1. If M is a maximal ideal of D and d = (d_{1}, …, d_{n}) ∈ D^{n}, let Q_{k} be the ideal of K[x_{1}, …, x_{n}] generated by (x_{1} − d_{1}), …, (x_{k} − d_{k}). Then a chain of primes of length n + 1 of Int_{D}^{n}(A) is given by Q_{0} = (0), Q_{k} = Q_{k} ∩ Int_{D}^{n}(A), for k = 1, …, n and Q_{n+1} = P_{d} = {f ∈ Int_{D}^{n}(A) | f(d_{1}, …, d_{n}) ∈ P }, where P is a maximal ideal of the image of d under Int_{D}^{n}(A). □

5.5 Remark. If D is a Noetherian domain of characteristic 0, finitely generated as a ℤ-algebra, such as, for instance, the ring of integers O_{K} in a number field, then no maximal ideal of Int_{D}^{n}(A) lies over (0) of D. This holds for A = D by [3, Prop. XI.3.4.], and carries over to Int_{D}^{n}(A), since Int_{D}^{n}(A) ⊆ Int(D^{n}). Every prime ideal P of Int_{D}^{n}(A) coming from a prime ideal of K[x_{1}, …, x_{n}] is contained in a maximal ideal Q of Int(D^{n}) lying over a maximal ideal M of D, and Q ∩ Int_{D}^{n}(A) then properly contains P.

5.6 Remark. Even if there are no maximal ideals lying over (0), maximal chains of primes of Int_{D}^{n}(A) are not necessarily of length n + 1. For instance, a maximal ideal of Int_{D}^{n}(A) may well have height 1 if it is of the form (MÂ)_{a} for a = (a_{1}, …, a_{n}) ∈ Â^{n} with a_{1}, …, a_{n} algebraically independent over K.

6. Integer-valued polynomials on matrix algebras

Theorem 4.6 characterizes the spectrum of the ring Int_{D}(A), provided we know the images of elements of M -adic completions of A under Int_{D}(A). We will now determine these images in the case A = M_{n}(D). Note that all the technical hypotheses in this section are certainly satisfied if D = O_{K} is the ring of integers in a number field.

6.1 Fact. [10, Lemma 2.2] Let D be a domain and f(x) = g(x)/d with g ∈ D[x], d ∈ D \ {0}. Then f ∈ Int_{D}(M_{n}(D)) if and only if g is divisible modulo dD[x] by all monic polynomials in D[x] of degree n.

6.2 Proposition*. Let D be a domain such that the intersection of the maximal ideals of D of finite index is (0), and f(x) = g(x)/d with g ∈ D[x], d ∈ D \ {0}. Then f ∈ Int_{D}(M_{n}(D)) if and only if g is divisible modulo dD[x] by all monic irreducible polynomials in D[x] of degree n.

Proof. In view of Fact 6.1, it suffices to show for every d ∈ D \ {0} and h ∈ D[x] monic of degree n, that there exists k ∈ D[x] monic of degree n, irreducible in D[x] and congruent to h mod dD[x]. We may choose a maximal ideal P of finite index with d ∉ P, and use Chinese remainder theorem on the coefficients of h to find k ∈ D[x], monic of degree n, congruent to h mod dD[x] and irreducible in (D/P)[x]. □

We are now able to identify a polynomially dense subset of M_{n}(D) consisting of companion matrices. They are often easier to work with than general matrices, because their characteristic polynomial is also their minimal polynomial.

6.3 Theorem*. Let C_{n} be the set of companion matrices of monic polynomials of degree n in D[x] and I_{n} ⊆ C_{n} the subset of companion matrices of irreducible polynomials. If D is any domain,

Int_{D}(M_{n}(D)) = Int_{D}(C_{n}, M_{n}(D)).

If D is a domain such that the intersection of the maximal ideals of finite index is (0), such as, for instance, a Dedekind domain with infinitely many maximal ideals of finite index, then

Int_{D}(M_{n}(D)) = Int_{D}(I_{n}, M_{n}(D)).

Proof. Let f ∈ Int_{D}(C_{n}, M_{n}(D)), f(x) = g(x)/d with g ∈ D[x], d ∈ D. Since g maps every C ∈ C_{n} to M_{n}(dD), g is divisible mod dD[x] by every monic polynomial in D[x] of degree n. (This is so because f is still the minimal polynomial of its companion matrix when everything is viewed in D/dD.) By Fact 6.1, f ∈ Int_{D}(M_{n}(D)). This shows Int_{D}(C_{n}, M_{n}(D)) ⊆ Int_{D}(M_{n}(D)). The reverse inclusion is trivial. The argument for I_{n} is similar, using Prop. 6.2. □

6.4 Theorem. Let D be a domain and C ∈ M_{n}(D). Let

Int(A)(C) = {f(C) | f ∈ Int_{D}(M_{n}(D))} and D[C] = {f(C) | f ∈ D[x]}.

Then Int(A)(C) = D[C].

Proof. Consider f ∈ Int_{D}(M_{n}(D)); f(x) = g(x)/d with g ∈ D[x] and d ∈ D \ {0}. We know that g is divisible modulo dD[x] by every monic polynomial in D[x] of degree n. Dividing g by χ_{C}, the characteristic polynomial of C, we get

g(x) = q(x)χ_{C}(x) + dr(x)

with q, r ∈ D[x] and we see that that f(C) = r(C). Thus Int(A)(C) ⊆ D[C]. The reverse inclusion is clear, since D[x] ⊆ Int_{D}(M_{n}(D)). □

6.5 Definition. A local domain is called analytically irreducible if its completion is also a domain.

6.6 Lemma. Let M be a maximal ideal of finite index in a domain D. Then the following are equivalent

(1) ⋂_{n=1}^{∞} M^{n} = (0) and D_{M} is analytically irreducible

(2) for every non-zero d ∈ D, cancellation of d is uniformly M -adically continuous.

Proof. (1 ⇒ 2) We have to show: for every non-zero d ∈ D, for every m ∈ ℕ there exists k ∈ ℕ such that for all c ∈ D: dc ∈ M^{k} implies c ∈ M^{m}. Indirectly, suppose there exists d ∈ D \ {0} and m ∈ ℕ, such that for every k ∈ ℕ there is some c_{k} ∈ D with dc_{k} ∈ M^{k} and c_{k} ∉ M^{m}. Since D̂ is compact and satisfies first countability axiom, (c_{k}) has a convergent subsequence. Its limit c ∈ D̂ satisfies c ∉ M^{m}, and hence c ≠ 0, and also for all k, dc ∈ M^{k}, which implies dc = 0. We have shown the existence of zero-divisors in D̂.

(2 ⇒ 1) is easy. □

6.7 Theorem. Let D be a domain, M a maximal ideal of finite index of D such that ⋂_{n=1}^{∞} M^{n} = (0) and D_{M} is analytically irreducible. Let D̂ be the M -adic completion of D, C ∈ M_{n}(D̂), and Int_{D}(M_{n}(D))(C) the image of C under Int_{D}(M_{n}(D)). Then

Int_{D}(M_{n}(D))(C) ⊆ D̂[C].

Proof. Let f ∈ Int_{D}(M_{n}(D)), f(x) = g(x)/d with g ∈ D[x], d ∈ D. For every m ∈ ℕ let k_{m} ∈ ℕ such that for all c ∈ D, cd ∈ M^{k_{m}} implies c ∈ M^{m}. Let E_{1} = (e_{ij}^{(1)}) and E_{2} = (e_{ij}^{(2)}) be matrices in M_{n}(D) with characteristic polynomials χ_{1} and χ_{2}. Then g(x) = q_{i}(x)χ_{i}(x) + dr_{i}(x) with q_{i}, r_{i} ∈ D[x] for i = 1, 2. If e_{ij}^{(1)} ≡ e_{ij}^{(2)} mod M^{k_{m}}, then dr_{1} ≡ dr_{2} mod M^{k_{m}}D[x], and therefore r_{1} ≡ r_{2} mod M^{m}D[x]. We can therefore M -adically approximate C by matrices C_{i} with f(C_{i}) = s_{i}(C_{i}) with s_{i} ∈ D[x], deg s_{i} < n, such that the s_{i} converge towards a polynomial s ∈ D̂[x] with deg s < n and f(C) = s(C). □

6.8 Corollary. Let D be a Dedekind domain, M a maximal ideal of finite index, D̂ the M -adic completion of D, and C ∈ M_{n}(D̂). Then

Int_{D}(M_{n}(D))(C) ⊆ D̂[C].

7. Integer-valued polynomials with matrix coefficients

While we have been investigating the ring Int_{D}(M_{n}(D)) of polynomials in K[x] mapping matrices in M_{n}(D) to matrices in M_{n}(D), Werner [13] has been studying the set, let’s call it Int_{D}[M_{n}(D)] with square brackets, of polynomials with coefficients in the non-commutative ring M_{n}(K) mapping matrices in M_{n}(D) to matrices in M_{n}(D). Without substitution homomorphism, it is not a priori clear that this set is closed under multiplication, but Werner [13] has shown that it is, and so Int_{D}[M_{n}(D)] is actually a ring between (M_{n}(D))[x] and (M_{n}(K))[x].

Also in [13], Werner proves that every ideal of Int_{D}[M_{n}(D)] can be generated by elements of K[x]. Using the idea of his proof, one can show more: Int_{D}[M_{n}(D)] is isomorphic to the algebra of n×n matrices over Int_{D}(M_{n}(D)). Since every prime ideal of a matrix ring is just the set of matrices with entries in a prime ideal of the ring, we get a description of the spectrum of Int_{D}[M_{n}(D)] as a byproduct of our description of the spectrum of Int_{D}(M_{n}(D)) in the previous section. We recall the definition of prime ideal for non-commutative rings:

7.1 Definition. We call a two-sided ideal P ≠R of a (not necessarily commutative) ring with identity R a prime ideal, if, for all ideals A, B of R,

AB ⊆ P ⟹ A ⊆ P or B ⊆ P,

or equivalently, if, for all a, b ∈ R

aRb ⊆ P ⟹ a ∈ P or b ∈ P.

For commutative R this is equivalent to the (in general stronger) condition: for all a, b ∈ R,

ab ∈ P ⟹ a ∈ P or b ∈ P.

7.2 Theorem. Let D be a domain with quotient field K, and

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)] ⊆ (M_{n}(K))[x] with its isomorphic image in M_{n}(K[x]) under

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))).

Proof. Note that K[x] is embedded in M_{n}(K[x]) as the subring of scalar matrices g(x)I_{n}, and in M_{n}(K)[x] as the subring of polynomials g(x) whose coefficients are scalar matrices rI_{n}, with r ∈ K. Clearly, Int_{D}[M_{n}(D)] ∩ K[x] = Int_{D}(M_{n}(D)).

Let C = (c_{ij}(x)) ∈ Int_{D}[M_{n}(D)] ⊆ M_{n}(K[x]). Let e_{ij} be the matrix in M_{n}(D) with 1 in position (i, j) and zeros elsewhere; then e_{ij}Ce_{kl} has c_{jk}(x) in position (i, l) and zeros elsewhere. Also, e_{ij}Ce_{kl} ∈ Int_{D}[M_{n}(D)], since Int_{D}[M_{n}(D)] is a ring containing M_{n}(D). So ∑_{i=1}^{n} e_{ij}Ce_{ki} = c_{jk}(x)I_{n} ∈ Int_{D}[M_{n}(D)]. Therefore c_{jk}(x)∈K[x]∩Int_{D}[M_{n}(D)]=Int_{D}(M_{n}(D)) for all (j, k), and hence Int_{D}[M_{n}(D)]⊆ M_{n}(Int_{D}(M_{n}(D))).

Conversely, if f ∈ Int_{D}(M_{n}(D)) then f(x)I_{n} ∈ Int_{D}[M_{n}(D)]. Therefore, e_{ik}f(x)I_{n}e_{kl}, the matrix containing f(x) in position (i, l) and zeros elsewhere, is in Int_{D}[M_{n}(D)], for arbitrary (i, l). By summing matrices of this kind we see that M_{n}(Int_{D}(M_{n}(D))) ⊆ Int_{D}[M_{n}(D)]. □

For any ring R with identity the ideals of R are in bijective correspondence with the ideals of M_{n}(R) by I ↦ M_{n}(I), and restriction to prime ideals gives a bijection between the spectrum of R and the spectrum of M_{n}(R). So we conclude:

7.3 Corollary. Let Int_{D}(M_{n}(D)) and Int_{D}[M_{n}(D)] as in the preceding theorem. Under the identification of Int_{D}[M_{n}(D)] with its isomorphic image in M_{n}(K[x]),

(1) The two-sided 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)).

(2) The two-sided prime ideals of Int_{D}[M_{n}(D)] are precisely the sets of the form M_{n}(P), where P is a prime ideal of Int_{D}(M_{n}(D)).

Acknowledgments. The author wishes to thank J.-L. Chabert for helpful hints, in particular Lemma 4.2 (iv), and P.-J. Cahen for improving Lemma 3.1.

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

frisch@TUGraz.at

Corrigendum for “INTEGER -VALUED POLYNOMIALS ON ALGEBRAS [J. Algebra 373 (2013) 414-425]”, appeared in J. Algebra 412 (2014) p 282

                    Sophie Frisch frisch@TUGraz.at

In this paper in J. Algebra 373 (2013) 414-425, in Prop. 6.2 and Thm. 6.3, instead of “a domain with zero Jacobson radical” we need the stronger assumption “a domain such that the intersection of all maximal ideals of finite index is zero”.