Contemporary Mathematics

Wedderburn Polynomials over Division Rings

T. Y. Lam and André Leroy

Abstract. A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K. Special cases of such polynomials include, for instance, the minimal polynomials (over the center F = Z(K)) of elements of K that are algebraic over F. In this note, we give a survey on some of our ongoing work on the structure theory of Wedderburn polynomials. Throughout the note, we work in the general setting of an Ore skew polynomial ring K[t, S, D].

§1. Introduction

The main object of this paper is the study of a class of polynomials over a division ring K, which we call Wedderburn polynomials (or simply W-polynomials). Roughly speaking, a W-polynomial over K is one which has “enough zeros” in K. (For a more precise definition, see (3.1).) In the case when K is a field, Wpolynomials are simply those of the form (t − c_{1}) ⋯ (t − c_{n}), where c_{1}, …, c_{n} are distinct elements of K. In the general case of a division ring K, a W-polynomial still has the form (t − c_{1}) ⋯ (t − c_{n}), although the c_{i} ’s need no longer be distinct. And even if the c_{i} ’s are distinct, (t − c_{1}) ⋯ (t − c_{n}) need not be a W-polynomial. The recognition of a W-polynomial, for instance, is already a very interesting problem over a division ring K.

The early work of Wedderburn [We] (ca. 1921) showed that, if a ∈ K is an algebraic element over the center F of K, then the minimal polynomial of a over F (in the usual sense) is a W-polynomial in K[t] (and in particular splits completely over K). This classical result of Wedderburn has led to much research on K[t], and has found important applications to the study of subgroups of the multiplicative group K^{∗}, central simple algebras of low degrees and crossed product algebras, PI-theory, Vandermonde matrices, Hilbert 90 Theorems, and the theory of ordered division rings, etc. For some literature along these lines, see [Al], [HR], [Ja_{3}], [La], [LL_{1}]-[LL_{3}], [Ro_{1}]-[Ro_{3}], [Se], [RS], ... , etc.

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1991 Mathematics Subject Classification. Primary 16D40, 16E20, 16L30; Secondary 16D70, 16E10, 16G30.

Some of the results in this note were presented by the first author in a lecture (with the same title) at the National Conference on Algebra VII, at Beijing Normal University in October, 1999.

We owe a debt of gratitude to Professor Zhang Yingbo, who kindly provided a Chinese translation of an English version of this article.

Our definition of W-polynomials was directly inspired by the afore-mentioned work of Wedderburn, although our W-polynomials will have coefficients in K, instead of in F. These W-polynomials have different properties and characterizations, and seem to be quite basic in working with the polynomial theory over K. We have recently initiated a systematic investigation of the theory of W-polynomials. In this note, we’ll give a synopsis of some of the results obtained so far. These include, for instance:

• A theorem on the left-right symmetry of W-polynomials;

• Several criteria for the factors and products of W-polynomials to be Wpolynomials;

• Some “Rank Theorems” describing the behavior of the rank of algebraic sets with respect to union, intersection, and certain other transformations;

• Applications of quadratic W-polynomials to the study of certain “metroequations” in division rings; and

• A study of the relationship between the set of W-polynomials and the set of algebraic subsets of K, from the viewpoint of complete modular lattices.

Following Ore [Or], we work in the setting of skew polynomials (rather than just ordinary polynomials) over the division ring K. This added degree of generality is definitely worthwhile considering that skew polynomials have become increasingly important with the growing interests in quantized structures and noncommutative geometry. The basic mechanism of skew polynomials is recalled in §2, where we also set up the terminology and general framework for the paper. Wedderburn polynomials are introduced in §3, and the synopsis of the main results on Wpolynomials is given in §§ 4-8.

Details of the research reported in this note (including the proofs of the results indicated above) will appear in the authors’ forthcoming papers [LL_{4}] and [LL_{5}].

§2. Skew Polynomial Rings and Algebraic Sets in K

To work with skew polynomials, we start with a triple (K, S, D), where K is a division ring, S is a ring endomorphism of K, and D is an S-derivation on K. (The latter means that D is an additive endomorphism of K such that D(ab) = S(a)D(b) + D(a)b, ∀a, b ∈ K.) In this general setting, we can form K[t, S, D], the Ore skew polynomial ring consisting of polynomials of the form ∑ b_{i}t^{i} (b_{i} ∈ K) which are added in the usual way and multiplied according to the rule

(2.1)  tb = S(b)t + D(b) ( ∀ b ∈ K).

In case (S, D) = (I, 0) (we’ll refer to this as the “classical case”), K[t, S, D] boils down to the usual polynomial ring K[t] with a central indeterminate t. Throughout this paper, we’ll write R := K[t, S, D]. It is easy to check that R admits an euclidean algorithm for right division, so R is a principal left ideal domain.

In working with R, it is important to be able to “evaluate” a polynomial f(t) = ∑ b_{i}t^{i} at any scalar a ∈ K, that is, to define f(a). Following our earlier work [LL_{1}], we take f(a) to be ∑ b_{i}N_{i}(a), where the “i th power function” N_{i} is defined inductively by

(2.2)  N_{0}(a) = 1, and N_{i}(a) = S(N_{i−1}(a))a + D(N_{i−1}(a)), ∀ a ∈ K.

That this gives the “right” definition of f(a) is seen from the validity of the Remainder Theorem [LL_{1}: (2.4)]: there is a unique q ∈ R such that

f(t) = q(t)(t − a) + f(a).

From this, it follows immediately that f(a) = 0 iff t − a is a right factor of f(t).

Another remarkable fact about evaluating skew polynomials at scalars is the “Product Formula” [LL_{1}: (2.7)] for evaluating f = gh at any a ∈ K:

(2.3)  (gh)(a) = \{ \begin{matrix}0 & if h(a) = 0, \\ g(a^{h(a)})h(a) & if h(a) ≠ 0. \end{matrix}

Here, for any c ∈ K^{∗}, a^{c} denotes S(c)ac^{−1} + D(c)c^{−1}, which is called the (S, D)- conjugate of a (by c). With this general conjugation notation, it is easy to verify by a direct calculation that

(2.4)  (a^{c})^{d} = a^{dc} for any c, d ∈ K^{∗}.

From this, it follows readily that (S, D)-conjugacy is an equivalence relation. In the following, we shall write

(2.5)  Δ^{S,D}(a) := { a^{c} : c ∈ K^{∗} };

this is called the (S, D)-conjugacy class of a. All such classes form a partition of K. For instance, Δ^{S,D}(0) is the set of all logarithmic derivatives {D(c)c^{−1} : c ∈ K^{∗}}. And, in the classical case, Δ^{I,0}(a) is just the “usual” conjugacy class

Δ(a) = {cac^{−1} : c ∈ K^{∗}}.

Next, we introduce a basic notation for this paper. For f ∈ R, let

(2.6)  V (f) := {a ∈ K : f(a) = 0}.

We’ll say that a set Δ ⊆ K is algebraic (or, more precisely, (S, D)-algebraic) if Δ ⊆ V (f) for some nonzero f ∈ R. In this case, the set of polynomials vanishing on Δ forms a nonzero left ideal in R. The monic generator of this left ideal is called the minimal polynomial of Δ; we denote it by f_{Δ}. The degree of f_{Δ} is called the rank of the algebraic set Δ ; we denote it by rk (Δ). According to the Remainder Theorem, f_{Δ} is just the (monic) “llcm” (least left common multiple) of the linear polynomials {t − a : a ∈ Δ}. As in [La: Lemma 5], it is easy to see that f_{Δ} has always the form (t − c_{1}) ⋯ (t − c_{n}) where each c_{i} is (S, D)-conjugate to some element of Δ.

Of course, all of the above was inspired in part by classical algebraic geometry. Going a little further, we get a theory of polynomial dependence (or P-dependence for short) for the elements of K. By definition, an element d ∈ K is P-dependent on an algebraic set Δ if g(d) = 0 for every g ∈ R vanishing on Δ. We see easily that the set of elements P-dependent on Δ is precisely V (f_{Δ}), which we shall henceforth call the “P-closure” of Δ and denote by \overline{Δ}. As in [La], we can also define P-independence and the notion of a P-basis for an algebraic set Δ in a natural manner. The cardinality of a P-basis for Δ is just rk (Δ). If {a_{1}, …, a_{n}} is a P-basis of Δ (where n = deg(f_{Δ})), then f_{Δ} is in fact the llcm of the linear polynomials {t − a_{i} : 1 ≤ i ≤ n}. We refer the reader to [La] for the rudiments of the theory of P-dependence. (Although this theory was developed in [La] in the case D = 0, it holds word-for-word also in the (S, D)-case.)

§3. The Class W of Wedderburn Polynomials

We now come to the principal object of the paper.

Definition 3.1. A polynomial f ∈ R is said to be a Wedderburn polynomial (or simply a W-polynomial) if it is the minimal polynomial of some algebraic set in K, or, equivalently, if f = f_{V (f)}. The set of W-polynomials in R will be denoted throughout this paper by the symbol W.

Some easy characterizations of W-polynomials are given in the following result.

Proposition 3.2. For a monic polynomial f ∈ R of degree n, the following are equivalent:

(1) f is a W-polynomial;

(2) rk (V (f)) = n (“f has enough zeros”);

(3) For any p ∈ R, V (f) ⊆ V (p) ⟹ p ∈ R · f.

To motivate the study of W-polynomials, we mention the following basic examples.

Examples 3.3.

(1) The constant polynomial 1 ∈ R is a W-polynomial, as it is the minimal polynomial of the empty (algebraic) set ∅.

(2) For any a ∈ K, t − a is a W-polynomial, as it is the minimal polynomial of the singleton (algebraic) set {a}.

(3) An easy argument shows that a monic quadratic polynomial f ∈ R is a W-polynomial iff |V (f)| ≥ 2. Alternatively, the quadratic W-polynomials are those of the form (t − b^{b−a})(t − a), where a ≠ b in K. For instance, over the quaternions, with (S, D) = (I, 0), f := t^{2} + 1 ∈ W since V (f) contains i and j; but g := (t − j)(t − i) ∉ W, since V (g) = {i}.

(4) If K is a field and (S, D) = (I, 0), the algebraic sets are precisely the finite subsets of K. It follows that, in this case, the W-polynomials are precisely the completely split polynomials (t − a_{1}) ⋯ (t − a_{n}) where the a_{i} ’s are distinct elements in K.

(5) Let (S, D) = (I, 0), and let a ∈ K be any element that is algebraic over F = Z(K), with minimal polynomial f(t) ∈ F [t] (in the usual field-theoretic sense). By Wedderburn’s Theorem in [We], the conjugacy class Δ := Δ(a) is algebraic (in our sense), with f_{Δ} = f(t). Therefore, f ∈ F [t] is a W-polynomial over K. From this example, we see that the W-polynomials introduced in (3.1) are a generalization of the minimal polynomials of algebraic elements studied by Wedderburn in [We].

One reason the family W commands our interest is that there are many possible ways to characterize the Wedderburn polynomials. The characterizations in (3.2) above are only the simpler ones that follow directly from the definitions. To close this section, we’ll mention below a few other more substantial characterizations.

Consider for the moment the case where K is a field and (S, D) = (I, 0). For a monic polynomial f = t^{n} + b_{n−1}t^{n−1} + ⋯ + b_{0} ∈ R, we have the following well-known “companion matrix”:

(3.4)  C(f) := ( \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{0} & −b_{1} & −b_{2} & ⋯ & −b_{n−1} \end{matrix} ) ,

whose characteristic and minimal polynomials are both f. For this companion matrix to be diagonalizable, the classical criterion is that f has n distinct (characteristic) roots in K. By (3.3)(4) above, this is also the criterion for f to be a Wedderburn polynomial! Through this example, we see that W-polynomials ought to be related to diagonalization questions in general.

In the case when R = K[t, S, D], we can consider the R-module R/Rf as a left K-vector space, with the K-basis {1, t, …, t^{n−1}}. With respect to this basis, C(f) is the matrix corresponding to left multiplication by t. However, this action of t is only a pseudo-linear transformation (in the sense that t(kv) = S(k)t(v) + D(k)v, for k ∈ K and v ∈ R/Rf). Upon a change of K-basis on R/Rf, the matrix of a pseudo-linear transformation will change by an “(S, D)-conjugation” (analogous to the (S, D)-conjugation of elements defined in §2). Using such a generalized notion of conjugation on matrices, we have the following characterization result for W -polynomials, which, in particular, extends the classical results on the diagonalizability of the companion matrix C(f).

Theorem 3.5. For any polynomial f = t^{n} + b_{n−1}t^{n−1} + ⋯ + b_{0} ∈ R, the following are equivalent:

(1) f is a product of linear factors, and R/Rf is a semisimple R-module.

(2) The companion matrix C(f) in (3.4) can be (S, D)-conjugated to a diagonal matrix.

(3) There exists an invertible Vandermonde matrix^{1} V = V (c_{1}, …, c_{n}) over K such that C(f)V = S(V) diag(c_{1}, …, c_{n}) + D(V) . (Here, S and D operate on matrices in the obvious way.)

(4) deg(f) = ∑_{j=1}^{r} dim_{C_{j}} E(f, a_{j}).

In this theorem, we have slipped in another characterization (4), which requires some notational explanations. The elements a_{1}, …, a_{r} there are representatives of the (S, D)-conjugacy classes of K in which f has roots, and the “exponential space” E(f, a_{j}) is defined to be

{0} ∪ {x ∈ K^{∗} : f(a_{j}^{x}) = 0}.

This is a right vector space over the division subring

C_{j} := {0} ∪ {c ∈ K^{∗} : a_{j}^{c} = a_{j}} (the “(S, D)-centralizer” of a_{j}).

1For any given triple (K, S, D), the general Vandermonde matrix V (c_{1}, …, c_{n}) is defined to be the n × n matrix whose (i + 1) st row is (N_{i}(c_{1}), …, N_{i}(c_{n})), where N_{i} is the i th power function (with respect to (S, D)) defined in (2.2).

For any f, it is true that ∑_{j=1}^{r} dim_{C_{j}} E(f, a_{j}) ≤ deg(f); (4) of the theorem above tells us that equality here is a criterion for f to be a Wedderburn polynomial.

§4. Left-Right Symmetry of W-Polynomials

According to the discussions in §2, W-polynomials are the (monic) llcm’s of polynomials of the form t − a (a ∈ K). In other words, a monic polynomial f belongs to W iff the left ideal Rf can be expressed as an intersection of the form ⋂ R(t−a), where a ranges over some subset of K. From this description, it would appear that W is a family dependent on the left structure of the ring R. Our first result on W is that it is in fact a left-right symmetric family.

To state this symmetry result more formally, let us introduce the following notations. Let W^{ℓ} be the family of monic polynomials f ∈ R such that Rf is an (arbitrary) intersection of principal left ideals of the form R(t − a) (a ∈ K), and let W_{0}^{ℓ} be the family of monic polynomials f ∈ R such that Rf = R(t − a_{1}) ∩ ⋯ ∩ R(t − a_{n}), where a_{i} ∈ K and n = deg(f). By using right (instead of left) principal ideals, we can similarly define two families of monic polynomials W^{r} and W_{0}^{r}. The symmetry result [LL_{4}: (4.5)] states the following:

Theorem 4.1. W = W^{ℓ} = W_{0}^{ℓ} = W_{0}^{r} = W^{r}.

The first two equalities here are already covered by our previous discussions, so the gist of the theorem is in the last two equalities. We refer the reader to [LL_{4}] for the proof of (4.1), but let us give here some indication of how one could achieve a passage from the left structure of R to its right structure. The following result, in fact, gives a constructive description of the inclusion W_{0}^{ℓ} ⊆ W_{0}^{r} in (4.1).

Theorem 4.2. Suppose f ∈ R is monic of degree n, and Rf has a representation as R(t − a_{1}) ∩ ⋯ ∩ R(t − a_{n}). (In other words, f is the minimal polynomial of the P-independent set {a_{1}, …, a_{n}}.) If h_{i} is the minimal polynomial of {a_{1}, …, a_{i−1}, a_{i+1}, …, a_{n}}, then, for b_{i} := a_{i}^{h_{i}(a_{i})} (1 ≤ i ≤ n), we have a representation fR = ⋂_{i=1}^{n}(t − b_{i})R.

A constructive description of the reverse inclusion W_{0}^{r} ⊆ W_{0}^{ℓ} can be given “similarly”.

The Symmetry Theorem (4.1) serves to show that the family W of Wedderburn polynomials is sufficiently intrinsic to the ring R to be characterizable in terms of either its left structure or its right structure. In the case when S is an automorphism of K, this is not too surprising, and was in fact known in a more general form to Ore (see [Or: Ch. II, Th. 4]). But if S is just an endomorphism of K, the situation is different. In case S(K) ≠ K, R is not a principal right ideal domain (or even a right Ore domain), so there is an apparent disparity between the left and the right structures of R. Here, one seems to have no à priori reason to expect a symmetry result such as (4.1).

What comes to one’s rescue is the notion of a semifir due to P. M. Cohn. A semifir is a domain in which every finitely generated left ideal is free of a unique rank — and this is known to be a left-right symmetric notion (see [Co_{2}]). Since our ring R = K[t, S, D] is a principal left ideal domain, it is a (left and and hence right) semifir. From this fact, it turns out that one has enough symmetry information on R to prove Theorem 4.1. We refer the reader to [LL_{4}] for the details of the proof.

An interesting byproduct of these considerations is a theory of left roots of polynomials in R. For f ∈ R, we have defined earlier

(4.3)  V (f) = {a ∈ K : f(a) = 0} = {a ∈ K : f ∈ R(t − a)};

such elements a should be called the right roots of f. We can correspondingly define

(4.4)  V′(f) = {b ∈ K : f ∈ (t − b)R};

such b ’s may then be called the left roots of f. This opens the way to many more interesting facts. For instance, in analogy to (3.2)(3), one can prove the following characterization of W-polynomial in terms of left root sets:

Proposition 4.5. For any monic f, we have f ∈ W iff, ∀ q ∈ R, V′(f) ⊆ V′(q) implies q ∈ fR.

In general, however, the right and left root sets V (f) and V′(f) behave rather differently, especially in the case when S(K) ≠ K. In this case, the theorem below says that V′(f) is situated in a very special way with respect to the coset partition of K modulo the additive subgroup S(K). This result has no right root analogue, and thus represents a new feature of the left root theory.

Theorem 4.6. For any nonzero polynomial f, either all left roots of f lie in a single S(K)-coset of K, or no two left roots lie in the same S(K)-coset. If f is monic, then necessarily all left roots of f lie in a single S(K)-coset of K.

§5. Factor Theorem, Product Theorem, and the Φ-Transform

Wedderburn polynomials turn out to have very tractable properties under multiplication and division. In this section, we’ll state several theorems on the factors and products of W-polynomials.

First we have to clarify what exactly is meant by the word “factor” in this paper. Throughout the sequel, we’ll say that a polynomial p is a factor of another polynomial f if f = p_{1}pp_{2} for some p_{1}, p_{2} ∈ R. Right and left factors of f have their usual meanings, and these are, of course, particular kinds of factors in our sense. The main result about factors of W-polynomials is the following (cf. [Or: Ch. II, Th. 3]):

Factor Theorem 5.1. For any monic f ∈ R, the following are equivalent: (1) f is a W-polynomial ;

(2) f splits completely, and every monic factor of f is a W-polynomial ;

(3) f splits completely, and every monic quadratic factor of f is a W-polynomial.

Next, we’ll discuss the situation of products. In the classical case when (S, D) = (I, 0) and K is a field, we see already that the product of two W-polynomials need not be a W-polynomial. In fact, if g, h ∈ W in this case, we’ll have gh ∈ W iff V (g) ∩ V (h) = ∅. One must then look for generalizations of this statement to the case of R = K[t, S, D] (for any triple (K, S, D)).

To solve this problem, we need the tool of a certain “Φ-transform”. In the Product Formula (2.3) for evaluating gh at a, we first encountered the expression a^{h(a)} (in case h(a) ≠ 0); similar expressions also occurred in the statement of Theorem (4.2). This led us to the following formal definition.

Definition 5.2. For h ∈ R = K[t, S, D], we define the “Φ -transform” (associated to h)

(5.3)  Φ_{h} : K \ V (h) ⟶ K

by Φ_{h}(x) = x^{h(x)}, whenever h(x) ≠ 0. (We do not attempt to define Φ_{h} on V (h).) Note that Φ_{h} always preserves the (S, D)-conjugacy class of its argument x.

Examples 5.4.

(1) Say h is a nonzero constant (polynomial) c ∈ K^{∗}. Then, Φ_{c}(x) = x^{c} for all x ∈ K, so Φ_{c} is defined on all of K and is exactly (S, D)-conjugation by the element c. (In particular, Φ_{1} is just the identity map on K.) In view of this example, we can think of the Φ-transform as a kind of generalization of (S, D)-conjugation.

(2) In the case D = 0 and h = t, Φ_{t}(a) = a^{a} = S(a)aa^{−1} = S(a) for any a ≠ 0. Here, Φ_{t} is the restriction of S to K^{∗}.

With the aid of the Φ-transform, one can prove the following result which gives various criteria for the product of two W-polynomials to be another W-polynomial.

Product Theorem 5.5. For f = gh ∈ R where g, h are monic, the following are equivalent:

(1) f ∈ W;

(2) g, h ∈ W, and 1 ∈ Rg + hR;

(3) g, h ∈ W, and V (g) ⊆ im(Φ_{h});

(4) g, h ∈ W, and, ∀ a ∈ V (g), b ∈ V′(h), we have (t − a)(t − b) ∈ W.

The formulation (4) of a W-polynomial criterion in terms of quadratics is particularly nice since (3.3)(3) makes it quite easy to recognize quadratic W-polynomials; see also Theorem (8.2) below.

§6. Three Rank Theorems

In this section, we state a few theorems concerning the ranks of algebraic sets. Recall that, for an algebraic set Δ, rk (Δ) is the degree of its minimal polynomial f_{Δ}, and the P-closure \overline{Δ} of Δ is (or may be taken to be) V (f_{Δ}). We’ll say that an algebraic set Δ is full if \overline{Δ} = Δ.

The first rank formula below gives a relation between the ranks of the union and intersection of algebraic sets. The formula is obviously prompted by the well-known dimension equation in the theory of finite-dimensional vector spaces. However, the use of the P-closures in the intersection \overline{Δ} ∩ \overline{Γ} below adds a subtle element to the formula. (Of course, the formula would look simpler if we restrict ourselves to full algebraic sets. But this would be an unnecessary sacrifice of generality.)

First Rank Theorem 6.1. For any two algebraic sets Δ and Γ, we have

rk (Δ) + rk (Γ) = rk (Δ ∪ Γ) + rk (\overline{Δ} ∩ \overline{Γ}).

In particular, rk (Δ ∪ Γ) = rk (Δ) + rk (Γ) iff \overline{Δ} ∩ \overline{Γ} = ∅.

We move on now to the Second Rank Theorem, which deals with the change of the rank of an algebraic set under a Φ-transform, say Φ_{h}. One basic example to keep in mind is when h is a constant polynomial c ∈ K^{∗}. In this case, Φ_{h} is (S, D)-conjugation by c (see (5.4)(1)), which is easily seen to be rank-preserving on algebraic sets. In the general case, a Φ-transform Φ_{h} will be rank-decreasing, and the precise result is as follows.

Second Rank Theorem 6.2. Let h ∈ R, and Δ ⊆ K be an algebraic set disjoint from V (h). Then

rk (Φ_{h}(Δ)) = rk (Δ) − rk (\overline{Δ} ∩ V (h)).

In particular, rk (Φ_{h}(Δ)) = rk (Δ) iff \overline{Δ} ∩ V (h) = ∅.

A powerful application of the rank formula for Φ_{h} is the following result giving a natural bound on the rank of the zero set of a product of two arbitrary polynomials.

Third Rank Theorem 6.3. For f = gh ∈ R \ {0}, we have

rk (V (f)) ≤ rk (V (g)) + rk (V (h)).

In spite of its simplicity, (6.3) is not an easy result. Note, for instance, the following interesting application of it to W-polynomials. Suppose g, h above are monic, of degrees r and s respectively. If we are given f := gh ∈ W, then by (3.2), rk (V (f)) = deg(f) = r + s, and (6.3) gives

rk (V (g)) + rk (V (h)) ≥ r + s = deg(g) + deg(h).

This implies that rk (V (g)) = r and rk (V (h)) = s, and hence (again by (3.2)) g, h ∈ W. This conclusion amounts essentially to the implication (1) ⟹ (2) in the Factor Theorem (5.1).

§7. A Duality Between Two Complete Modular Lattices

The considerations in the previous sections lead us quickly to the construction of several lattices. There is only one more piece missing in the whole puzzle. This is given by the following proposition, where “rgcd” is used as an abbreviation for “right greatest common divisor”.

Proposition 7.1. The intersection of any nonempty family of full algebraic sets {Δ_{j} : j ∈ J} is also a full algebraic set, with minimal polynomial given by rgcd {f_{Δ_{j}} : j ∈ J}.

Remark 7.2. The fact that ⋂_{j} Δ_{j} has minimal polynomial rgcd{f_{Δ_{j}} : j ∈ J} is generally not true if the algebraic sets Δ_{j} are not all full. For instance, in the division ring K of real quaternions, Δ = {i} is full and Γ = {j, k} is not full. The rgcd of f_{Δ} = t − i and f_{Γ} = t^{2} + 1 is t − i. But Δ ∩ Γ = ∅ has minimal polynomial 1.

Now consider the poset F = F(K, S, D) of all full algebraic sets in K (with respect to (S, D)), where the partial ordering is given by inclusion:

(7.3)  Δ ≤ Γ ⟺ Δ ⊆ Γ (for Δ, Γ ∈ F ).

This poset is a lattice, with Δ ∧ Γ given by Δ ∩ Γ (which lies in F by (7.1)), and with Δ ∨ Γ given by \overline{Δ ∪ Γ}. (The union Δ ∪ Γ is algebraic, but may not be full, as the case |Δ| = |Γ| = 1 already shows.) Note that the lattice F has a smallest element, given by the empty set ∅ (see (3.3)(1)).

If K itself is not (S, D)-algebraic, F will not have a largest element. In this case, we can define an augmented lattice F^{∗} by adjoining to F a largest element. If K is (S, D)-algebraic, then it is the largest element in F, and we can simply define F^{∗} to be F.

To get a lattice in the context of (skew) polynomials, we consider the set W = W(K, S, D) of all W-polynomials, partially ordered by:

(7.4) f ≤ h ⟺ Rf ⊆ Rh ⟺ h is a right divisor of f (for f, h ∈ W).

The poset W is again a lattice: for f, h ∈ W as above, f ∨ h is given by rgcd(f, h) (this being a W-polynomial by the Factor Theorem), and f ∧ h is given by llcm(f, h) (this being a W-polynomial since it is the minimal polynomial of V (f) ∪ V (h)). The lattice W has a largest element, given by 1 ∈ F, and it will have a smallest element iff K happens to be (S, D)-algebraic. (The smallest element in the latter case is the minimal polynomial of K itself.) In analogy with the case of full algebraic sets, we can introduce an augmented W-polynomial lattice W^{∗}, which is defined to be W if K is (S, D)-algebraic, and W adjoined with the polynomial 0 otherwise (this being smaller than all other W-polynomials).

Once we have set up the augmented lattices F^{∗} and W^{∗} as above, we can use (6.1), (7.1), etc. to prove the following duality result.

Theorem 7.5. For a fixed triple (K, S, D), we have the following:

(1) F^{∗} and W^{∗} are both complete modular lattices. The maps Δ ↦ f_{Δ} and f ↦ V (f) (extended in the obvious way) define mutually inverse lattice dualities between F^{∗} and W^{∗}.

(2) The map “rk” (extended in the obvious way) is the “dimension function” on the modular lattice F^{∗} (in the sense of lattice theory), and the degree map “deg” is the “dual dimension function” on the modular lattice W^{∗}.

(3) The minimal elements (the so-called atoms) of the lattice F^{∗} are the singleton subsets of K, and the maximal elements of the lattice W are the monic linear polynomials in R.

(4) For f ≤ h ∈ W, the “interval” [f, h] in the lattice W is isomorphic to the lattice of all R-submodules of Rh/Rf.

In our proof of (1) of this theorem in [LL_{5}], the modularity of F^{∗} is not proved directly, but is rather deduced from the modularity of W^{∗}. It is, therefore, of interest to record the following statement, which essentially amounts to the modular law for F^{∗}.

Proposition 7.6. Let Γ, Π and Δ be algebraic sets, where Γ, Π are full, and Δ ⊆ Γ. If x ∈ Γ is P-dependent on Π ∪ Δ , then it is already P-dependent on the smaller set (Γ ∩ Π) ∪ Δ.

§8. The (S, D)-Metro Equation

In the theory of division rings, the study of the equation ax − xb = c has had a long history, going back to the work of R. E. Johnson [Jo] and N. Jacobson [Ja_{2}]. By a slight abuse of the terminology of P. M. Cohn ([Co_{1}], [Co_{3}]), we shall call ax − xb = c a “metro equation” over K. It turns out that the notion of metro equations bears a close relationship to that of Wedderburn polynomials.

First, we generalize the classical metro equation notion to our (S, D)-setting. This is not difficult: for a, b, c ∈ K, let us call

(8.1)  ax − S(x)b − D(x) = c

the (S, D)-metro equation (associated with a, b, c). (Of course, when (S, D) = (I, 0), (8.1) boils down to the ordinary metro equation ax − xb = c.) In the case c = 0, (8.1) has an obvious solution x = 0, so in the following, we’ll assume c ≠ 0 whenever (8.1) is considered. The following result gives the precise relationship between (8.1) and quadratic Wedderburn polynomials, in the general (S, D)-setting.

Theorem 8.2. For any a, b ∈ K and c ∈ K^{∗}, the following are equivalent: (1) The (S, D)-metro equation ax − S(x)b − D(x) = c has a solution in K;

(2) (t − b^{c})(t − a) ∈ W;

(3) 1 ∈ R · (t − b^{c}) + (t − a) · R.

Here, (2) ⟺ (3) follows directly from (5.5), so what is new in (8.2) is the equivalence (1) ⟺ (2). Just for the sake of illustration, let us give the proof for this equivalence in the “classical case”, namely, when (S, D) = (I, 0). This will, in fact, serve as the one and only proof given in this entire paper.

What we are trying to prove here is that ax − xb = c has a solution in K iff the quadratic

f(t) := (t − b^{c})(t − a) = t^{2} − (a + b^{c})t + b^{c}a ∈ K[t]

(where now b^{c} := cbc^{−1}) is a W-polynomial. For the latter to be true, we need to have a (right) root of f in K that is different from a (see (3.3)(3)). Write such a root (if it exists) in the form a − cx^{−1} (where x ∈ K^{∗}). For this to be a right root of f, we need:

0 = f(a − cx^{−1}) = (a − cx^{−1})^{2} − (a + b^{c})(a − cx^{−1}) + b^{c}a
= (a − cx^{−1} − a − b^{c})(a − cx^{−1}) + b^{c}a
= (−cx^{−1} − b^{c})(a − cx^{−1}) + b^{c}a
= c(−x^{−1}a + bx^{−1} + x^{−1}cx^{−1}),

that is, x^{−1}a − bx^{−1} = x^{−1}cx^{−1}. Multiplying this equation by x from the left and from the right, we get it down to ax − xb = c. Therefore, finding a root for f other than a amounts exactly to solving the metro equation ax − xb = c!

Example 8.3. Let K be the division hull of the Weyl algebra ℂ⟨u, v⟩, with the relation uv − vu = 1. Choosing a = b = u, we see that the metro equation ax − xb = 1 has a solution x = v. Thus, it follows from (8.2) that f(t) := (t − u)^{2} is a W-polynomial over K. (Two roots for f are u and u − v^{−1}, according to the calculation above.) This example is interesting, as it shows that a Wedderburn polynomial may very well have repeated linear factors.

Theorem (8.2) has some nice applications to the solvability of the metro equation (8.1) in case the (S, D)-conjugacy class of b is an algebraic set. We close by stating the following consequence of (8.2).

Proposition 8.4. Suppose the (S, D)-conjugacy class Δ := Δ^{S,D}(b) is algebraic, and let a ∈ K \ Δ, c ∈ K^{∗}. Then (t − b^{c})(t − a) ∈ W, and the (S, D)-metro equation (8.1) has a unique solution.

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Department of Mathematics, University of California, Berkeley, CA 94720

E-mail address: lam@math.berkeley.edu

Department of Mathematics, Université d’Artois, 62307 Lens Cedex, France

E-mail address: leroy@euler.univ-artois.fr