in: S. Chapman (ed.), Arithmetical Properties of Commutative Rings and Monoids (Chapel Hill Conf. 2003), Lect. Notes in Pure and Appl. Math. vol. 241, Chapman&Hall CRC 2005, pp 253–259.

Polynomial Separation of Points in Algebras

SOPHIE FRISCH Institut für Mathematik, Technische Universität Graz, A-8010 Graz, Austria email: frisch@blah.math.tu-graz.ac.at

ABSTRACT. We show that for a wide variety of domains, including all Dedekind rings with finite residue fields, it is possible to separate any two algebraic elements a, b of an algebra over the quotient field by integer-valued polynomials (i.e. to map a and b to 0 and 1, respectively, with a polynomial in K[x] that maps every element of D to an element of D), provided only that the minimal polynomials of a and b in K[x] are co-prime (which is obviously necessary).

In contrast to this, it is impossible to separate a, b ∈ D by a n × n-integer-matrix-valued polynomial (a polynomial in K[x] that maps every n×n matrix over D to a matrix with entries in D), except in the trivial case where a−b is a unit of D. (This is despite the fact that the ring of n × n-integer-matrix-valued polynomials for any fixed n is non-trivial whenever the ring of integer-valued polynomials is non-trivial.) 2000 Math. Subj. Classification: Primary 13F20; Secondary 13B25, 11C08, 15A36, 16B99.

1 INTRODUCTION

Definition 1.1. If D is a domain with quotient field K, the ring of integer-valued polynomials of D is

Int(D) = {f ∈ K[x] | ∀d ∈ D: f(d) ∈ D}.

Rings of integer-valued polynomials have proved remarkably well suited for interpolation purposes.

Definition 1.2. A domain D is called interpolation domain, if for all n ∈ ℕ, for all a_{1}, …, a_{n} ∈ D (distinct) and all b_{1}, …, b_{n} ∈ D there exists f ∈ Int(D) with f(a_{i}) = b_{i} for 1 ≤ i ≤ n.

Interpolation domains have been classified both among Noetherian domains and among Prüfer domains in [3] (see also Corollary 2.4). In particular, every Dedekind domain all of whose residue fields are finite is an interpolation domain.

Equivalent to being an interpolation domain is the following point separation property: for any a, b ∈ D with a ≠ b there exists a polynomial f ∈ Int(D) with

f(a) = 0 and f(b) = 1. 1

In this note we explore two possible avenues for extending this point separation property to D-algebras and K-algebras. If D is a domain for which separation of points in D by polynomials in Int(D) is possible, we ask ourselves

(1) Can elements in an arbitrary K-algebra (for instance the ring of n×n matrices with entries in K) be separated by polynomials in Int(D)?

(2) Can elements of D be separated by polynomials in K[x] that map a given D-algebra (for instance the ring of n × n matrices with entries in D) to itself? It turns out that the answer to (1) is yes, within reason (the minimal poly-

nomials of the elements to be separated have to be co-prime in K[x]); but (2) may fail spectacularly for D-algebras with zero-divisors. We will show that it is impossible to separate points in D by polynomials in K[x] that map M_{n}(D) (the D-algebra of n × n matrices with entries in D) to itself, except in trivial cases.

2 SEPARATION OF POINTS IN AN ALGEBRA

The basic connections between separation of points, co-maximality of null-ideals, and interpolation hold in great generality, so we state them for functions - even though we are only concerned with polynomials in this paper.

Throughout this paper, every ring or algebra (commutative or not) is assumed to have an identity element (and every ring homomorphism to satisfy φ(1) = 1). D and K stand for a domain and its quotient field; R denotes a ring (possibly non-commutative, possibly with zero-divisors).

Notation and Conventions. Let R be a ring and S a set. We say that a ring F is a ring of functions from S to R if F comes equipped with a ring homomorphism ψ: F → R^{S}(= ∏_{s∈S} R) assigning to each element f ∈ F a function (which we also call f, by abuse of notation).

If (F, ψ) is a ring of functions from S to R and s ∈ S, we call

N_{F}(s) = {f ∈ F | f(s) = 0}

the null-ideal of s in F.

We say that we can separate points in S by functions from F if for all s, t ∈ S with s ≠ t there exists an f ∈ F with f(s) = 0 and f(t) = 1.

We say that we can interpolate arbitrary functions from S to R by functions in F if for all n ∈ ℕ, for all s_{1}, …, s_{n} ∈ S (distinct) and all r_{1}, …, r_{n} ∈ R there exists some f ∈ F with f(s_{i}) = r_{i} for 1 ≤ i ≤ n.

Lemma 2.1. Let (F, ψ) be a ring of functions from S to R.

(i) For s, t ∈ S, there exists an f ∈ F with f(s) = 0 and f(t) = 1 if and only if

N_{F}(s) + N_{F}(t) = F.

(ii) To be able to interpolate arbitrary functions from S to R by functions in F it is necessary and, if F is an R-algebra and ψ an R-algebra homomorphism, also sufficient, to be able to separate points in S by functions in F.

Proof. (i) is easy; and, ad (ii), separation of points is clearly necessary for interpolation. If F is an R-algebra then, given s_{1}, …, s_{n} ∈ S which we can mutually separate, we have functions f_{ij} ∈ F with f_{ij}(s_{i}) = 1 and f_{ij}(s_{j}) = 0, which we multiply to get a Lagrange interpolation function f_{i} ∈ F with f_{i}(s_{i}) = 1 and f_{i}(s_{j}) = 0 for j ≠ i. Then we can use R-linear combinations of the f_{i} to interpolate. □

Proposition 2.2. Let D be a domain with quotient field K. D is an interpolation domain if and only if it has the following property: whenever a_{1}, a_{2} are algebraic elements of a K-algebra A, whose minimal polynomials in K[x] are co-prime, there exists a polynomial p ∈ Int(D) with p(a_{1}) = 0 and p(a_{2}) = 1.

Proof. Assume D is an interpolation domain. Given a_{1}, a_{2} with co-prime minimal polynomials f_{1}, f_{2} ∈ K[x], let c_{1}, c_{2} be non-zero elements of D such that c_{i}f_{i} = g_{i} ∈ D[x]. There exist polynomials p_{i} ∈ D[x] and a non-zero constant d ∈ D such that

g_{1}(x)p_{1}(x) + g_{2}(x)p_{2}(x) = d.

Let h(x) ∈ Int(D) with h(0) = 1 and h(d) = 0, and set

p(x) = h(g_{2}(x)p_{2}(x)) = h(d − g_{1}(x)p_{1}(x)),

then p(a_{1}) = 0 and p(a_{2}) = 1.

Conversely, the point separation property for K-algebras specialized to A = K implies separation of points in D by polynomials in Int(D) which by Lemma 2.1 (ii) implies interpolation of arbitrary functions from D to D by polynomials in Int(D). □

If D is a Dedekind domain with finite residue fields, the construction of the point separating integer-valued polynomial h in the preceding proof can be made fairly explicit, cf. [4].

Corollary 2.3. A domain D with quotient field K is an interpolation domain if and only if for any f_{1}, f_{2} ∈ K[x] with

K[x]f_{1}(x) + K[x]f_{2}(x) = K[x]

it follows that

(K[x]f_{1}(x) ∩ Int(D)) + (K[x]f_{2}(x) ∩ Int(D)) = Int(D).

Proof. In view of Lemma 2.1 and Proposition 2.2, all that remains to show is that every two non-constant monic polynomials in K[x] occur as minimal polynomials of two elements of the same K-algebra. This can be seen by looking at the companion matrices of the polynomials in question, bringing the smaller matrix up to the size of the larger matrix by padding with zeros at the right and bottom. □

Recalling the characterization of interpolation domains among Noetherian domains from [3] (Theorem 2.4.), we obtain:

Corollary 2.4. Let D be a Noetherian domain with quotient field K. Then the following are equivalent:

(1) D is an interpolation domain (2) whenever a_{1}, a_{2} are algebraic elements of a K-algebra A whose minimal poly-

nomials in K[x] are co-prime, there exists a polynomial p ∈ Int(D) with p(a_{1}) = 0 and p(a_{2}) = 1

(3) whenever I, J are ideals of K[x] with I + J = K[x] then

(I ∩ Int(D)) + (J ∩ Int(D)) = Int(D)

(4) D is one-dimensional with finite residue fields and locally unibranched (the last condition meaning that the integral closure of every localization at a prime ideal is again local).

3 INTEGER-MATRIX-VALUED POLYNOMIALS

Definition 3.1. We denote by Mint_{n}(D) the ring of polynomials in K[x] that map n × n matrices with entries in D to matrices with entries in D:

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

Clearly,

K[x] ⊇ Int(D) ⊇ Mint_{2}(D) ⊇ Mint_{3}(D) ⊇ … ⊇ D[x],

and, as Lemma 3.4 below shows, ⋂_{n=1}^{∞} Mint_{n}(D) = D[x].

Example 3.2. One may ask if Mint_{n}(D) is not often trivial, in the sense of Mint_{n}(D) = D[x]. The answer is, for Noetherian D at least: not more often than Int(D) is trivial. (This can be shown by combining the proof of Theorem I.3.14 of [2] with Lemma 3.4 below). It follows immediately from Lemma 3.4 that Mint_{n}(D) ≠ D[x] (for any n ≥ 1) whenever D has a proper principal ideal of finite index. For example,

\frac{x^{6} + x^{5} + x^{3} + x^{2}}{2} is in Mint_{2}(ℤ) \ Mint_{3}(ℤ).

(The residue class of x^{6} + x^{5} + x^{3} + x^{2} in ℤ_{2}[x] is the least common multiple of all monic polynomials in ℤ_{2}[x] of degree 2.)

When investigating Mint_{n}(D), it is useful to know in what cases the null ideal of a matrix A ∈ M_{n}(R) in the polynomial ring R[x] is principal, not just for R = D, but also for residue class rings R = D/I. If D is a domain then N_{D[x]}(A) = {f ∈ D[x] | f(A) = 0} is principal for every matrix A ∈ M_{n}(D) if and only if D is integrally closed [5]. But even for the most mundane commutative ring R we know that the null ideals in R[x] of some matrices are principal. (Thanks to Gabriel Picavet for pointing this out):

Lemma 3.3. Let R be a commutative ring, f ∈ R[x] a monic polynomial and A ∈ M_{n}(R) the companion matrix of f. Then N_{R[x]}(A) = f(x)R[x].

Proof. By a theorem of McCoy ([1], Theorem 7.31) the null ideal in R[x] of a matrix A ∈ M_{n}(R) is (F_{n} :_{R[x]} F_{n−1}), where F_{k} denotes the ideal generated in R[x] by the k × k minors of xI_{n} − A (I_{n} the n × n identity matrix). For any A, F_{n} = c_{A}(x)R[x] (c_{A} the characteristic polynomial of A); and if A is the companion matrix of a polynomial, then one of the (n − 1) × (n − 1) minors of xI_{n} − A is 1, such that in this case the null ideal of A is c_{A}(x)R[x] = f(x)R[x]. □

Lemma 3.4. Let D be a domain and f(x) = g(x)/c, g ∈ D[x], c ∈ D \ {0}. Then f ∈ Mint_{n}(D) if and only if g is divisible modulo cD[x] by all monic polynomials in D[x] of degree n.

Proof. Suppose f ∈ Mint_{n}(D). Let ḡ denote the residue class of g in (D/cD)[x]. Then ḡ maps every matrix in M_{n}(D/cD) to the zero-matrix. In particular, this holds for all companion matrices of monic polynomials of degree n. Therfore ḡ is divisible in (D/cD)[x] by all monic polynomials of degree n.

Conversely, this condition suffices for f ∈ Mint_{n}(D), because it means that g is divisible modulo cD[x] by the characteristic polynomial of every matrix in M_{n}(D), and therefore maps every matrix in M_{n}(D) to a matrix in M_{n}(cD). □

We are now ready to show that the rings Mint_{n}(D) are as unsuitable as can be for separation of points. For n > 1, Mint_{n}(D) cannot even separate elements a, b ∈ D, except in the trivial case when a − b is a unit and they can be separated by the polynomial (x − b)/(a − b) ∈ D[x].

Theorem 3.5. Let D be a domain and a, b ∈ D. If there exists a polynomial f ∈ Mint_{2}(D) with f(a) = 1 and f(b) = 0 then a − b is a unit of D.

Proof. By linear substitution we can reduce to the case a = 0. Let f ∈ Mint_{2}(D) with f(0) = 1 and f(b) = 0. We write f as f(x) = g(x)/c with g ∈ D[x] and c ≠ 0 in D. It follows that c = g(0) and g(x) = (x − b)h(x) with h ∈ D[x].

By Lemma 3.4, g is divisible modulo cD[x] by all monic polynomials in D[x] of degree 2, in particular by (x − b)x.

In (D/cD)[x] we have, for some k ∈ (D/cD)[x],

(x − b)xk(x) = ḡ(x) = (x − b)h̄(x)

and we can cancel (x − b) which, being monic, is certainly not a zero-divisor in (D/cD)[x]. This shows that h̄, the residue class of h in (D/cD)[x], is divisible by x. Therefore, in D[x], the constant coefficient of h is divisible by c, say h(0) = dc. Looking at the constant coefficient of g we have c = g(0) = −bh(0) = −bdc. Cancelling c ≠ 0 proves −b to be a unit of D. □

REFERENCES

[1] W. C. Brown, Matrices over Commutative Rings, vol. 169 of Pure and Applied Mathematics, Dekker, 1993.

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

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

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

[5] , Integrally closed domains, minimal polynomials, and null ideals of matrices, Comm. Algebra, 32 (2004), pp. 2015–2017.

Corrigendum. After publication I discovered a slight mistake in the proof of Corollary 2.3. The corollary itself is correct. Here’s how to fix the proof: Given two non-constant monic polynomials f and g in K[x], we want a K-algebra A that has two elements a and b whose minimal polynomials are f and g. The matrices given in the paper won’t work. What does work is: let C_{f} and C_{g} be the companion matrices of f and g, let A = M_{n}(K), with n = lcm(deg f, deg g) and let a and b be block diagonal matrices, such that the blocks of a are all equal to C_{f} and the blocks of b are all equal to C_{g}.