PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 181–186 DOI: 10.2298/PIM131217001A

CENTERS OF SKEW POLYNOMIAL RINGS

Waldo Arriagada and Hugo Ramírez

Abstract. We determine the center C(K[x; δ]) of the ring of skew polynomials K[x; δ], where K is a field and δ is a non-zero derivation over K. We prove that C(K[x; δ]) = ker δ, if δ is transcendental over K. On the contrary, if δ is algebraic over K, then C(K[x; δ]) = (ker δ)[η(x)]. The term η(x) is the minimal polynomial of δ over K.

1. The ring of skew polynomials

Let K be a field and let δ be a non-zero derivation on K, cf. [4, 13]. That is, a linear function δ : K → K such that δ(ab) = aδ(b) + δ(a)b for any a, b ∈ K. The ring of skew polynomials in x and coefficients in K is the set

K[x; δ] = { ∑_{i∈ℕ} x^{i}a_{i} : {a_{i}} ⊂ K has finite support} ,

endowed with the usual equality, addition and equipped with multiplication rule: ax = xa+δ(a), a ∈ K. Since their formal introduction in the 1930’s by Oystein Ore, skew polynomial rings and their iterated constructions have been further developed by N. Jacobson, S. A. Amitsur, P. M. Cohn, G. Cauchon, T. Y. Lam, A. Leroy, and J. Matczuk, and complete treatments can be found in the literature, cf. [5, 9, 10]. Computationally, such rings appear in the context of uncoupling and solving systems of linear differential and difference equations in closed form, cf. [1, 3, 12].

First, we notice that the multiplication rule and induction yield the relation:

(1.1)  ax^{n} = ∑_{k=0}^{n} ( {n \atop k}) x^{k}δ^{n−k} (a),  a ∈ K, n ⩾ 1.

In addition, the following properties are satisfied in the ring K[x; δ], see [2, 6, 8].

(1) The function deg : K[x; δ] ∖ {0} → ℕ defined by deg(f) = max{i : a_{i} ≠ 0}, where f(x) = ∑ _{i} x^{i}a_{i} ≠ 0 , is a degree function satisfying

deg(f ± g) ⩽ max{deg(f); deg(g)}

2010 Mathematics Subject Classification: 12E15, 12E10.

Key words and phrases: derivation, skew polynomial, center, ring, commutator.

Communicated by Žarko Mijajlović.

Further, deg(f · g) = deg(f) + deg(g), for f, g ∈ K[x; δ] ∖ {0}.

(2) K[x; δ] is an integral domain satisfying the right and left division algorithm. Hence, it is a principal right and left ideal domain.

(3) Consider the triple (A, a_{0}, σ), where A is a ring, a_{0} ∈ A and σ : K → A is a ring homomorphism. If

(1.2)  a_{0}σ(a) = σ(a)a_{0} + σ(δ(a)), a ∈ K,

then there exists a unique σ̄ : K[x; δ] → A with σ̄|_{K} = σ and σ̄(x) = a_{0} . Further, when such a σ̄ exists, σ̄(f(x)) = fσ̄(a_{0}), where fσ̄(a_{0}) ∈ A is obtained via substitution of each coefficient of f(x) by its image under σ̄, and x by a_{0}.

Corollary 1.1. The ring homomorphism σ : K → End(K, +), a ↦ σ(a), where σ(a) : b ↦ ba, extends as a unique σ̄ : K[x; δ] → End(K, +), where σ̄(x) = δ. Moreover, σ̄ is K-linear when End(K, +) is considered as a right K-vector space via the action fλ = f(σ(λ)).

Proof. To extend σ it suffices that identity (1.2) be fulfilled, with a_{0} = δ. But this is the case. Indeed, let a ∈ K; one has

[σ(a) ∘ δ + σ(δ(a))](b) = δ(b)a + bδ(a) = δ(ab) = (δ ∘ σ(a))(b), b ∈ K.

The extension σ̄ is K-linear since σ̄(fλ) = σ̄(f) ∘ σ̄(λ) = σ̄(f)(σ(λ)) = σ̄(f)λ. □

Let σ̄ : K[x; δ] → End(K, +) be the linear homomorphism of Corollary 1.1 and

K[δ] = Im(σ̄) = { ∑_{i=0}^{n} δ^{i}a_{i} : n ∈ ℕ,; a_{i} ∈ K} ,

where δ^{i}a_{i} = δ^{i} ∘σ(a_{i}). Then K[δ] is a subring of End(K, +) isomorphic to K[x; δ]. It is, moreover, a right vector space over K.

Lemma 1.1 (Jacobson–Bourbaki). Let A ∋ 1 be a subring of End(K, +) such that: (a) A is a right vector subspace of End(K, +) over K, (b) [A : K] = ν. Then k = {a ∈ K : σ(a) ∘ f = f ∘ σ(a), ∀f ∈ A} is a subfield of K such that [K : k] = ν. Further, A = End_{k}(K, +) as a vector space over k.

The next result is a standard consequence of the division algorithm in K[x; δ], cf. [9].

Proposition 1.1. Let f(x) ∈ K[x; δ] be of minimal degree deg(f), such that f(δ) = 0. Then [K[δ] : K] = deg(f). If g ∈ K[x; δ] satisfies g(δ) = 0, then f divides g.

2. Commutator

Let A be a ring. The commutator of a, b ∈ A is [a, b] = ab − ba. It is easy to see that the commutator satisfies the following properties:

(4) [·, ·] : A × A → A is bi-additive.

(5) For every a, b ∈ A the functions δ_{a}, δ_{b} : A → A defined by δ_{a} = [a, ·] and δ_{b} = [·, b] are derivations. They are called the internal derivations induced by a and b, respectively.

(6) Any ring homomorphism σ : A → B preserves the commutator, i.e.,

σ[a, b] = [σ(a), σ(b)], a, b ∈ A.

In the sequel we will denote κ = kerδ = {a ∈ K : δ(a) = 0}. (And hence, κ is a subfield of the field K.)

Proposition 2.1. In the ring A = K[x; δ] the following properties are fulfilled.

(a) [a, x^{n}] = ∑ _{k=0}^{n−1} \binom{n}{k}x^{k}δ^{n−k} (a), for each a ∈ K and n ⩾ 1. In particular, [a, x] = δ(a), and hence [a, x^{n}] = 0 for all a ∈ κ.

(b) [x^{n}a, x] = x^{n}[a, x] for all a ∈ K and n ∈ ℕ.

(c) The commutator [·, ·] : K[x; δ] × K[x; δ] → K[x; δ] is κ-bilinear.

Proof. (a) This is a consequence of (1.1):

[a, x^{n}] = ax^{n} − x^{n}a = ∑_{k=0}^{n} ( {n \atop k}) x^{k}δ^{n−k} (a) − x^{n}a = ∑_{k=0}^{n−1} ( {n \atop k}) x^{k}δ^{n−k} (a).

(b) [x^{n}a, x] = x^{n}ax − xx^{n}a = x^{n}ax − x^{n+1} a = x^{n}(ax − xa) = x^{n}[a, x].

(c) Let f ∈ K[x; δ] and let δ_{f} (·) = [·, f] be the internal derivation induced by f. Then for every a, λ ∈ κ and any positive integer i,

[λx^{i}a, f] = δ_{f} (λx^{i}a) = δ_{f} (λ)x^{i}a + λδ_{f} (x^{i}a) = [λ, f]x^{i}a + λ[x^{i}a, f] = λ[x^{i}a, f],

since [λ, f] = 0. Indeed, if f(x) = ∑ _{i=0}^{n} x^{i}a_{i}, then by (4),

[λ, ∑ x^{i}a_{i}] = ∑ δ_{λ}(x^{i}a_{i}) = ∑ (δ_{λ}(x^{i})a_{i} + x^{i}δ_{λ}(a_{i})) = ∑ ([λ, x^{i}]a_{i} + x^{i}[λ, a_{i}]),

and this last quantity vanishes by (a). Hence, [·, f] is homogeneous on monomials x^{i}a_{i} and by additivity it is homogeneous on polynomials. We prove in an analogous way that [f, ·] is homogeneous on polynomials. □

Corollary 2.1. Let f(x) = ∑ _{i=0}^{n} x^{i}a_{i} ∈ K[x; δ] and a ∈ K. Then

(a) [f(x), x] = ∑ _{i=0}^{n} x^{i}δ(a_{i}),

(b) [a, f(x)] = ∑ _{i=1}^{n} ∑ _{k=0}^{i−1} \binom{i}{k}x^{k}δ^{i−k} (a)a_{i} .

Proof. (a) [∑ x^{i}a_{i}, x] = ∑ x^{i}[a_{i}, x] = ∑ x^{i}δ(a_{i}).

(b) Since the commutator is bi-additive, we have

[a, f(x)] = [a, a_{0} + xa_{1} + ⋯+ x^{n}a_{n}] = [a, a_{0}] + [a, xa_{1}] + ⋯+ [a, x^{n}a_{n}]
= [a, x]a_{1} + ⋯+ [a, x^{n}]a_{n}.

The conclusion follows from Proposition 2.1(a). □

3. Results

The center of a ring A is the subring C(A) = {a ∈ A : [a, b] = 0, ∀b ∈ A}. The elements of C(A) are called central.

Lemma 3.1. Let a, b ∈ C(A) with b cancelable. If c ∈ A such that a = bc, then c is also central.

Proof. For any α ∈ A, 0 = αa − aα = αbc − bcα = b(αc − cα). □

The next lemma is a useful result that follows from Kummer’s theorem [7].

Lemma 3.2. [11] Let j be a nonzero natural number and p a prime number. Assume that j = j_{0} + j_{1}p + ⋯+ j_{r}p^{r} (j_{r} ≠ 0) is the p-adic decomposition of j. If \binom{j}{k} = 0 (mod p) for all k = 1, …, j − 1 then j_{0} = ⋯= j_{r−1} = 0, j_{r} = 1. That is, j = p^{r}.

The next theorem is the core result of the paper.

Theorem 3.1. If the derivation δ is transcendental over K, then C(K[x; δ]) = κ. On the contrary, if δ is algebraic over K, then there exist a positive integer N and a prime p such that ch(K) = p (where ch(K) is the characteristics of K) and the center C(K[x; δ]) = κ[η(x)], where

η(x) = xα_{1} + x^{p^{2}} α_{2} + ⋯+ x^{p^{N}} α_{i} ∈ kerδ,

is the minimal polynomial of δ over K. In this case we have also [K : κ] = p^{N}.

Proof. Let us assume that f(x) = ∑ _{i=0}^{m} x^{i}b_{i} ∈ K[x; δ], with b_{m} ≠ 0 and m = deg(f) ⩾ 1. Then f(x) is central if and only if f(x) commutes with x and with every a ∈ K. That is, [f(x), x] = [a, f(x)] = 0. By Corollary 2.1, these conditions are simultaneously equivalent to b_{i} ∈ κ for i = 0, …, m and

(3.1) δ(a)b_{1} + ⋯+ ∑_{k=0}^{i−1} ( {i \atop k}) x^{k}δ^{i−k} (a)b_{i} + ⋯+ ∑_{k=0}^{m−1} ( {m \atop k}) x^{k}δ^{m−k} (a)b_{m} = 0,

for every a ∈ K. Hence, if δ is transcendental over K, the coefficients b_{i}, i = 1, …, m vanish and f(x) = b_{0} ∈ κ, which proves the first statement of the theorem.

Let us suppose that δ is algebraic over K. Since (3.1) is a null polynomial, the coefficient of the power x^{i−1} is zero. That is:

δ( {i \atop i − 1} ) b_{i} + δ^{2}( {i + 1 \atop i − 1} ) b_{i+1} + ⋯+ δ^{m−i+1} ( {m \atop i − 1} ) b_{m} = 0, i = 1, …, m.

In particular, δb_{1} + δ^{2}b_{2} + ⋯+ δ^{m}b_{m} = 0 for i = 1, and δmb_{m} = 0 for i = m . Since δ ≠ 0, δmb_{m} = 0 ⇒ mb_{m} = 0 ⇔ (m · 1_{K})b_{m} = 0, where 1_{K} is the multiplicative unit of K. Hence, m · 1_{K} = 0 and then there exists a prime number p such that ch(K) = p. (In particular, ch(K) is nonzero). Further, if we define g(x) = f(x) − f(0), then g(x) is central and g(δ) = 0. Hence, there exists a monic polynomial η(x) ∈ K[x; δ] with η(0) = 0 and of minimal degree deg(η) = n ⩾ 1, such that η(δ) = 0. Write

(3.2)  η(x) = ∑_{i=1}^{n} x^{i}a_{i},

with a_{n} = 1. Inasmuch as δ(1) = 0, by Corollary 2.1(a) the bracket

[η(x), x] = ∑_{i=1}^{n−1} x^{i}δ(a_{i})

has a lower degree than η(x). Consider the homomorphism σ̄ of Corollary 1.1. We have σ̄[η(x), x] = [σ̄(η(x)), σ̄(x)] = [η(δ), δ] = [0, δ] = 0. Hence, [η(x), x] = 0. Similarly, for every a ∈ K and by Corollary 2.1(b), the commutator [a, η(x)] has a lower degree than η(x). Also, σ̄[a, η(x)] = [σ̄(a), σ̄(η(x))] = [σ(a), η(δ)] = [σ(a), 0] = 0. Hence, [a, η(x)] = 0, for every a ∈ K. Thus, η is central. On the other hand, we notice that an element a ∈ K is central if and only if a ∈ κ. (Indeed, [a, q(x)] = 0 if and only if δ(a) = 0, for q ∈ K[x; δ]). Hence, κ[η(x)] ⊂ C(K[x; δ]).

Conversely, we prove by induction on the degree of f(x) that if f(x) is central, with deg(f) ⩾ 1, then there exists g ∈ κ[x] such that f(x) = g(η(x)). Indeed, f central ⇒ f(δ) − f(0) = 0 ⇒ η(x) divides f(x) − f(0) ⇔ f(x) − f(0) = η(x)f_{1}(x). By Lemma 3.1, f_{1} is central. If f_{1}(x) ∈ κ, then the polynomial g(x) = f(0) + f_{1}(x) ∈ κ[x] satisfies f(x) = g(η(x)). By minimality, if deg(f_{1}) ⩾ 1, and since deg(f_{1}) < deg(f), we have f_{1}(x) = g_{1}(η(x)) where g_{1} ∈ κ[x]. This means f(x) = f(0) + η(x)g_{1}(η(x)). Hence if g(x) = f(0) + xg_{1}(x) ∈ κ[x], we have f(x) = g(η(x)). This proves that C(κ[x; δ]) = κ[η(x)].

Now, let us determine the explicit form of the minimal polynomial (3.2). Since η(x) is central its coefficients satisfy δ\binom{i}{i−1}a_{i} +δ^{2} \binom{i+1}{i−1}a_{i+1} +⋯ +δ^{n−i+1} \binom{n}{i−1} = 0, for i = 1, …, n. That is, δ is root of the polynomials

f_{i}(x) = x( {i \atop i − 1} ) a_{i} + x^{2}( {i + 1 \atop i − 1} ) a_{i+1} + ⋯+ x^{n−i+1} ( {n \atop i − 1} ) ,

for i = 1, …, n. Notice that if i ⩾ 2 then 1 ⩽ deg(f_{i}) ⩽ n − 1. Then by minimality, the coefficients of f_{i}, with 2 ⩽ i ⩽ n, vanish identically. These coefficients can be arranged in rows:

\binom{2}{1}a_{2} \binom{3}{1}a_{3} ⋯ \binom{j}{1}a_{j} ⋯ \binom{n}{1},  i = 2
\binom{3}{2}a_{3} ⋯ \binom{j}{2}a_{j} ⋯ \binom{n}{2},  i = 3
...  ⋮  ⋮  ⋮
\binom{j}{j−1}a_{j} ⋯ \binom{n}{j−1},  i = j
...  ⋮  ⋮
\binom{n}{n−1},  i = n.

Thus if a_{j} ≠ 0, for 2 ⩽ j ⩽ n, then \binom{j}{k} · 1_{K} = 0 for all k = 1, …, j − 1. Hence,

( {j \atop 1}) = ( {j \atop 2}) = ⋯= ( {j \atop j − 1} ) = 0 (mod p).

By Lemma 3.2, there exists a positive integer r_{j} such that j = p^{r_{j}} . Let us denote N = r_{n}. For l = 2, …, N define coefficients

α_{l} = \{ \begin{matrix}a_{i} & if l = r_{i} for some i, \\ 0 & otherwise. \end{matrix}

Then the minimal polynomial has the required form η(x) = xα_{1}+x^{p^{2}} α_{2}+⋯ +x^{p^{N}} , where α_{1} = a_{1} and α_{i} ∈ ker(δ).

Finally, we know that K[δ] is a subring of End(K, +). It is moreover a right vector subspace of End(K, +) over K. By Proposition 1.1, [K[δ] : K] = deg(η) = p^{N} .

Further,

{a ∈ K : σ(a)∘A = A∘σ(a), A ∈ K[δ]} = {a ∈ K : σ(a)∘δ = δ ∘σ(a)} = kerδ = κ.

By Lemma 1.1, [K : κ] = p^{N} and K[δ] = End_{κ}(K, +). The proof is complete. □

Acknowledgments. The authors are grateful to an unknown referee, who provided helpful suggestions and comments.

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Department of Applied Mathematics and Sciences (Received 23 07 2012) Khalifa University (Revised 28 08 2013) Al Zafranah, P.O. Box 127788 Abu Dhabi United Arab Emirates

waldo.arriagada@kustar.ac.ae

Instituto de Ciencias Físicas y Matemáticas Universidad Austral de Chile Casilla 567, Valdivia Chile

hramirez@uach.cl