Comm. Algebra 32(5) (2004) 2015-2017

INTEGRALLY CLOSED DOMAINS, MINIMAL POLYNOMIALS, AND NULL IDEALS OF MATRICES

SOPHIE FRISCH

Abstract. We show that every element of the integral closure D′ of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of J. Brewer and F. Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x]. It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A, N_{D}(A) = {f ∈ D[x] | f(A) = 0} is a principal ideal of D[x].

For a square matrix A with entries in a domain D, the null ideal of A in D[x] is N_{D}(A) = {f ∈ D[x] | f(A) = 0}. For integrally closed D, W.C. Brown [3] has shown that this ideal is always principal. Conversely, if D is a domain for which all null ideals of matrices are principal, then D is integrally closed. We will show this by demonstrating that every element of the integral closure D′ of a domain D occurs as a coefficient of a minimal polynomial of a matrix with entries in D. This also answers a question of Brewer and Richman [2], namely, if integrally closed domains are characterized by the fact that the minimal polynomial of every square matrix with entries in D is in D[x]. To put this question in context, we remind the reader of a related, but a priori stronger, property characterizing integrally closed commutative rings:

Fact. Let R be a commutative ring and T its total ring of quotients. Then R is integrally closed in T if and only if it has the following property:

whenever f, g, and h are monic polynomials in T [x] with f(x) = g(x)h(x) then f ∈ R[x] implies g ∈ R[x] and h ∈ R[x].

Proof. (⇐) Suppose R has the property and u ∈ T is integral over R. Let f ∈ R[x] be a monic polynomial with f(u) = 0 then f(x) = g(x)(x − u) for some monic g ∈ T [x], therefore (x − u) ∈ R[x] and u ∈ R.

(⇒) This is shown in Bourbaki [1, Chpt 5, §1.3, Prop. 11] by means of the splitting ring of g and h. If R is a domain, this direction also follows from the fact that R is an intersection of valuation rings contained in T and in each valuation ring we have Gauß’s Lemma C(f) = C(g) C(h). □

If D is an integrally closed domain, then the above fact guarantees that the minimal polynomial of every matrix with entries in D is in D[x], since after all the minimal polynomial is a monic factor of the characteristic polynomial.

2000 Mathematics Subject Classification. Primary 13B22, 15A21; Secondary 13B25, 12E05, 11C08, 11C20.

Key words and phrases. Integrally closed domain, minimal polynomial of a matrix, null ideal of a matrix, linear algebra over commutative rings, matrices over commutative rings.

Theorem. Let D be a domain and D′ its integral closure. Then every element of D′ occurs as a coefficient of a minimal polynomial of a matrix with entries in D.

Proof. Let K be the quotient field of D and u ∈ K integral over D. We use the expression “second-highest coefficient” to designate the coefficient of x^{n−1} in a polynomial of degree n > 0.

Let f_{1}(x) be a monic polynomial in D[x] with f_{1}(u) = 0, deg f_{1} ≥ 3 and secondhighest coefficient zero. (Given any monic f ∈ D[x] with f(u) = 0, we can set f_{1}(x) = f(x)(x^{2} − cx), where c is the second-highest coefficient of f.)

We write u as a fraction u = a/b with a, b ∈ D and set f_{2}(x) = f_{1}(x) + (bx − a). Then f_{2}(x) is another monic polynomial in D[x] with deg f_{2} ≥ 3, second-highest coefficient zero and f_{2}(u) = 0.

In K[x], f_{1}(x) = g(x)(x−u) for some monic polynomial g ∈ K[x] with deg g ≥ 2, and f_{2}(x) = (g(x)+b)(x−u). Note that the second-highest coefficient in both g(x) and g(x) + b is u.

Now let A_{i} be the companion matrix of f_{i} for i = 1, 2 and A the block-diagonal matrix with A_{1} and A_{2} on the main diagonal. Then the minimal polynomial h(x) of A is the least common multiple of f_{1} and f_{2} in K[x]. Since g(x) and g(x) + b are relatively prime, the minimal polynomial of A is

h(x) = g(x) (g(x) + b) (x − u).

We have arranged things so that the three monic factors g(x), g(x)+b and (x−u) of h(x) have second-highest coefficients u, u, and −u, respectively. Therefore the second-highest coefficient of h(x) is u. □

Corollary. Let D be a domain. D is integrally closed if and only if the minimal polynomial of every square matrix with entries in D is in D[x].

If D is a domain with quotient field K and A a square matrix with entries in D, then the following conditions are easily seen to be equivalent (cf. [3]):

i) the minimal polynomial of A, m_{A}(x) ∈ K[x], is in D[x]. ii) the null ideal of A in D[x], N_{D}(A) = {f ∈ D[x] | f(A) = 0}, is principal.

(To see (ii ⇒ i): if N_{D}(A) is principal, it must have a monic generator, since it contains the characteristic polynomial of A, which is monic.) This yields the following variant of our characterization of integral domains:

Variant of Corollary. Let D be a domain. D is integrally closed if and only if for every square matrix A with entries in D, the null ideal of A in D[x],

N_{D}(A) = {f ∈ D[x] | f(A) = 0}

is a principal ideal.

References

1. N. Bourbaki, Commutative Algebra, Chapters 1–7, Springer, Berlin, 1989.

2. J. Brewer and F. Richman, Weakly integrally closed domains: minimum polynomials of matrices, Comm. Algebra 28 (2000), 4735–4748.

3. W. C. Brown, Null Ideals and Spanning Ranks of Matrices, Comm. Algebra 26 (1998), 2401– 2417.

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

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