Fields Institute Communications Volume 00, 0000

Dubrovin valuation rings and orders in central simple algebras

Joachim Gräter

Universität Potsdam Institut für Mathematik Postfach 601553 14469 Potsdam F.R.G.

                              graeter@rz.uni-potsdam.de

Abstract. During the last 20 years the theory of noncommutative valuations and valuation rings has been developed by many authors for different reasons. Especially valuation rings in finite-dimensional division algebras as well as their matrix rings are now very well understood and they lead to various interesting applications. The main progress in the general theory has been made after Dubrovin introduced his new type of valuation rings which are now called Dubrovin valuation rings. In this paper some of the main theorems on total and Dubrovin valuation rings are given. We concentrate on results which are related to the general extension problem such as the conjugacy theorem, the defect theorem as well as the characterization of the integral closure of a central valuation ring in terms of integral Bezout orders in central simple algebras.

1 Introduction

In this paper we present a survey on noncommutative valuation rings in finitedimensional division algebras and more general in central simple algebras. No proofs are given but we will indicate where proofs can be found (a highly recommended reference is [22] where also a comprehensive bibliography is included). Before we go into details let us first review some basic notation. Throughout this paper, every ring is assumed to have an identity 1 which is inherited by subrings. By a division algebra we shall understand a not necessarily commutative ring in which every non-zero element has a multiplicative inverse. Furthermore, if R is a ring then we write Z(R) for the center and J(R) for the Jacobson radical of R. If R is simple, then Z(R) is a field and R is called central simple algebra if R is simple and finite-dimensional over Z(R). By Wedderburn’s theorem any central simple

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 12J20,16H05; Secondary 16K20,16L30.

algebra Q is isomorphic to a matrix ring M_{n}(D) over a suitable division algebra D finite-dimensional over its center. D is called the underlying division algebra and n the matrix size of Q.

Section 2 deals with the more classical part of invariant and total valuation rings in division algebras finite-dimensional over their centers. The results here are very similar to the commutative case, especially to the case of a finite commutative Galois extension, with the exception that a general extension theorem does not exist. For instance, if D is finite-dimensional over its center F then a valuation ring V of F need not be extendible to D. Moreover, there exist examples where no central valuation ring of D can be extended to the entire algebra, i.e., D has no proper valuation ring at all. However, the presence of at least one extension leads to many nice results which are discussed in section 2. But the reader should be aware that some results in section 2 are just special versions of more general theorems about Dubrovin valuation rings presented in section 3 and there do not exist separate proofs in the literature.

In 1982 N. I. Dubrovin introduced a new type of valuation rings with significantly better extension properties than the classical valuation rings mentioned above (cf. [9, 10]). These rings are called Dubrovin valuation rings and they are not only defined for division rings but also for simple artinian rings, especially for central simple algebras. For example, if V is a valuation ring of the center F of a central simple algebra Q then V can always be extended to a Dubrovin valuation ring B of Q and all extensions are conjugate. Furthermore, if V is discrete then all Dubrovin extensions are precisely the maximal orders over V. Thus, Dubrovin valuation rings can be considered as the noncommutative analogue of commutative valuation rings in the sense of Krull ([21]) and they possess many additional properties noncommutative valuation rings are expected to have. Our aim in section 3 is to present the general extension theory of Dubrovin valuation rings in central simple algebras. The most basic problems are that for a central valuation ring V the integral closure in Q is not a ring and that the number of all extensions of V to Q is infinite in general. Therefore, it is not clear for instance how to describe the elements of Q which are integral over V and how to formulate a noncommutative version of the so-called ′′Ostrowski theorem′′ (defect theorem) for Dubrovin valuation rings. We will explain how these difficulties can be overcome by introducing the intersection property of Dubrovin valuation rings and the extension number n_{V} of a central valuation ring V. It turns out that V can be extended to an integral Bezout order R which is unique up to inner automorphisms. Furthermore, R is the intersection of precisely n_{V} Dubrovin extensions of V and any integral element of Q lies in at least one conjugate of R. Finally, n_{V} leads to a naturally generalization of the defect theorem which can be stated as [Q : F] = e_{V} · f_{V} · n_{V}^{2} · p^{δ} where e_{V} is the ramification index, f_{V} the residue degree, p the residue characteristic, and δ a nonnegative integer.

2 Total valuation rings

In this section D always means a finite-dimensional division algebra with center F where [D : F] = n^{2}. A subring B of D is said to be a (total) valuation ring of D, if

x ∈ B or x^{−1} ∈ B for all nonzero x ∈ D

and B is called proper whenever B ≠ D. In the following every valuation ring is assumed to be proper. Furthermore, if B is in addition invariant under all inner automorphisms, i.e., xBx^{−1} = B for all nonzero x in D, then B is called an invariant valuation ring. The invariant valuation rings are precisely those which arise from classical valuations (cf. [31]), but in this paper we concentrate on valuation rings only and so we do not pursue this theme. At this point it is worth remarking that a valuation ring of a finite-dimensional division algebra need not be invariant as the following example shows.

Example. Let K be a commutative field and let L be a finite cyclic Galois extension of K with Galois group < σ >. Let W be a valuation ring of L and let

D = L[[x, σ]] = {a_{n}x^{n} + a_{n−1}x^{n−1} + … | a_{i} ∈ L}

be the division algebra of all formal Laurent series where multiplication is defined via x · a = σ(a) · x, a ∈ L. It is not hard to verify that

F := Z(D) = K[[x^{n}]] and [D : F] = n^{2},

and that B and B′ are valuation rings of D where

B′ = {a_{0} + a_{1}x + a_{2}x^{2} + … | a_{0}, a_{1}, a_{2}, … ∈ L},
B = {a_{0} + a_{1}x + a_{2}x^{2} + … | a_{0} ∈ W; a_{1}, a_{2}, … ∈ L}.

Furthermore,

J(B′) = {a_{1}x + a_{2}x^{2} + … | a_{1}, a_{2}, … ∈ L},
J(B) = {a_{0} + a_{1}x + a_{2}x^{2} + … | a_{0} ∈ J(W); a_{1}, a_{2}, … ∈ L},
and xBx^{−1} = {σ(a_{0}) + σ(a_{1})x + σ(a_{2})x^{2} + … | a_{0} ∈ W; a_{1}, a_{2}, … ∈ L},
= {a_{0} + a_{1}x + a_{2}x^{2} + … | a_{0} ∈ σ(W); a_{1}, a_{2}, … ∈ L}.

While B′ is invariant in any case, B is invariant if and only if σ(W) = W. This is typical, since any noninvariant valuation ring of a finite-dimensional division algebra is contained in an invariant overring (cf. [3]), and an immediate consequence of this is that any valuation ring is invariant whenever it is maximal (cf. [7]). A major tool in commutative valuation theory is the general approximation theorem and one might ask whether a noncommutative analogue can also be proved for total valuation rings. In [13] it is shown that for arbitrary division algebras some serious problems occur, however results very similar to the results in the commutative case can be obtained for finite-dimensional division algebras. Thus, let B_{1}, …, B_{k} be pairwise incomparable valuation rings of D and let B_{ij} be the least overring of B_{i} and B_{j} for any pair (i, j). If I_{i} is a right ideal of B_{i} and a_{i} ∈ D then there exists an x ∈ D such that

x ≡ a_{1} mod I_{1}, …, x ≡ a_{k} mod I_{k}

if I_{i}B_{ij} = I_{j}B_{ij} and a_{i} − a_{j} ∈ I_{i}B_{ij} for all i, j. In [13] the approximation theorem is not stated exactly this way but it is not hard to check that both versions are equivalent. Originally, the formulation above was introduced by P. Morandi in [25] in order to investigate similar questions for Dubrovin valuation rings. If B is a valuation ring of D then B∩F = V is a valuation ring of F and we say that B is an extension of V (to D). The main problem in noncommutative valuation theory is that a central valuation ring need not be extendible to D. For instance, the quaternion division algebra D = (\frac{−1,−1}{ℚ(\sqrt{3})}) does not have any valuation ring, i.e., no valuation ring of ℚ(\sqrt{3}) can be extended to D. A detailed discussion of this and also other division algebras with prescribed finite sets of valuation rings can be found in [6]. Under the special assumption that F is henselian with respect to V (for details about henselian valuation rings see [12]) an extension always exists which is in addition unique and therefore invariant. This result is an immediate consequence of the following theorem (cf. [33]).

Theorem 2.1 A valuation ring V of F can be extended to an invariant valuation ring of D if and only if V can be extended uniquely to any subfield K of D containing F.

If V cannot be extended to an invariant valuation ring then the valuation of F belonging to V has no prolongation to D and vice versa. But this does not mean that V has no extension at all. Before we discuss this general situation let us recall some basic facts from commutative valuation theory. If L is a finite commutative extension of F then V can always be extended to L. The number of all extensions is bounded by [L : F] and the integral closure of V in L is the intersection of all of them. Furthermore, if L is a Galois extension of F then all extensions are conjugate. The following theorem shows that similar statements also hold in the noncommutative case provided that there exists at least one prolongation (cf. [3]).

Theorem 2.2 If V is a valuation ring of F then V can be extended to D if and only if the integral closure of V in D is a subring of D. Furthermore, if V can be extended to D, then the following hold.

1. V has only a finite number B_{1}, …, B_{k} of extensions.

2. k ≤ n = \sqrt{[D : F]}.

3. B_{1} ∩ … ∩ B_{k} is the integral closure of V in D.

4. All extensions of V are conjugate, i.e., if B and B′ are extensions of V to D then B′ = aBa^{−1} for some nonzero a in D.

If one chooses K, L, and W in the example above such that W, σ(W), …, σ^{n−1}(W) are all distinct then B, xBx^{−1}, …, x^{n−1}Bx^{−(n−1)} are all different extensions of B ∩ F = V. This shows that the upper bound \sqrt{[D : F]} for k in Theorem 2.2 can be reached. In general, k is strictly smaller that n = \sqrt{[D : F]} but still a divisor. This follows immediately from the defect theorem as we will see later on. For every valuation ring B of D the Jacobson radical J(B) is the unique maximal ideal of B since all left ideals and all right ideals are totally ordered by inclusion. Furthermore, any overring B′ of B in D is a valuation ring too and J(B′) is a completely prime ideal of B such that the correspondence B′ ⟷ J(B′) leads to a bijection between the set of all overrings and the set of all completely prime ideals of B (an ideal P of B is called completely prime if a ∈ P or b ∈ P whenever a·b ∈ P for all a, b ∈ B). In addition, the correspondence B′ ⟷ B′ ∩F leads to a bijection between the set of all overrings of B and the set of all overrings of V = B ∩ F in F. An immediate consequence of this is, that any overring of B is invariant whenever B is invariant itself (but this is not true in a general division algebra which is not finite-dimensional over its center). Even though a noninvariant valuation ring does not have an associated valuation it has a value group and we can also define the corresponding ramification index. In order to do this we first recall that a fractional two-sided ideal I of B is a proper left and right B-submodule of D and that the set of all nonzero fractional ideals F(B) is closed under multiplication. We say that I ∈ F(B) is invertible if there exists J ∈ F(B) such that IJ = JI = B. It is not difficult to verify that the set Γ_{B} of all invertible fractional two-sided ideals is an ordered abelian group with respect to this multiplication and the inclusion as ordering and that Γ_{B} = {aB | a ∈ D^{∗} such that aB = Ba}. The value group Γ_{V} of V can be considered canonically as a subgroup of the value group Γ_{B} of B and we call e_{B} = [Γ_{B} : Γ_{V}] the ramification index of B as usual. Since all extensions of V to D are conjugate, e_{B} does not depend on the special choice of B but only on V and so we also write e_{V} instead of e_{B}. The residue ring \overline{B} := B/J(B) is a division algebra which is finite-dimensional over its center and we called \overline{B} the residue division algebra of B. As in the commutative case, \overline{V} := V/J(V) can be viewed as a subfield of Z(\overline{B}) where V = B ∩ F and f_{B} = [\overline{B} : \overline{V}] is called residue degree of B. Again, f_{B} only depends on V and not on the special choice of B and therefore we also write f_{V} instead of f_{B}. The center Z(\overline{B}) of \overline{B} is a normal field extension of \overline{V} and Gal(Z(\overline{B})/\overline{V}) is a homomorphic image of Γ_{B}/Γ_{V} where \overline{V} and Z(\overline{B}) do not coincide in general. Now, we are able to state the defect theorem for total valuation rings (cf. [35, 14]).

Theorem 2.3 Let V be a valuation ring of F having n_{V} extensions to D. Then

[D : F] = e_{V} · f_{V} · n_{V}^{2} · p^{δ}

where p is the characteristic of \overline{V} and δ a nonnegative integer.

This theorem has first been proved by Draxl (cf. [8]) for the case where V is henselian and later on Morandi (cf. [24]) used his result to show that the defect theorem also holds for arbitrary invariant valuation rings (i.e. n_{V} = 1) by means of the so-called Henselization of a valued division algebra which is defined as follows. Let F^{h} be the Henselization of F with respect to V and let V^{h} be the corresponding valuation ring of F^{h} extending V and satisfying Γ_{V} = Γ_{V^{h}} and \overline{V} = \overline{V^{h}}. The tensor product D^{h} := F^{h} ⊗_{F} D is a central simple F^{h}-algebra and D^{h} is a division algebra if and only if V has an invariant extension B to D. In this situation V^{h} has a unique extension B^{h} to D^{h} and in [24] it is shown that B^{h} ∩ D = B, Γ_{B^{h}} = Γ_{B}, and \overline{B^{h}} = \overline{B}, i.e.,

[Γ_{B^{h}} : Γ_{V^{h}}] · [\overline{B^{h}} : \overline{V^{h}}] = [Γ_{B} : Γ_{V}] · [\overline{B} : \overline{V}].

This proves the defect theorem for invariant valuation rings using Draxl’s theorem:

[D : F] = [D^{h} : F^{h}] = e_{V^{h}} · f_{V^{h}} · p^{δ} = e_{V} · f_{V} · p^{δ}.

To derive the general version of Theorem 2.3 one can pursue two different ways, which were considered originally to prove a corresponding result for Dubrovin valuation rings (cf. [35, 14]). The following is a rough sketch of the method used in [14]. If n_{V} > 1 then there exists an overring V′ of V in F having an invariant extension B′ to D such that [\overline{B′} : Z(\overline{B′})] < [D : F] and all extensions of V are subrings of B′ (cf. [3]). Using induction on [D : F] we see that the defect theorem holds for V/J(V′) and \overline{B′} by induction hypothesis and the fact that Z(\overline{B′}) is a normal extension of \overline{V′} (cf. [12]). Finally, the defect theorem holds for V′ and D since B′ is an invariant extension of V′. It remains to express the ramification index and residue degree of V in terms of the ramification indices and residue degrees of V′ and V/J(V′). But this is not obvious and one needs some additional information about the connection between the ramification index of V′ and the number of extensions of V/J(V′) to Z(\overline{B′}). For details see [14]. In [35] Henselization methods were used and the following result was obtained:

Theorem 2.4 Let V be a valuation ring of F having n_{V} extensions to D. Then

F^{h} ⊗_{F} D ≅ M_{n_{V}} (D^{h})

where D^{h} is the underlying division algebra of F^{h}⊗_{F} D. Furthermore, if B^{h} denotes the unique extension of V^{h} to D^{h} then

[Γ_{B^{h}} : Γ_{V^{h}}] = e_{V} and [\overline{B^{h}} : \overline{V^{h}}] = f_{V} .

Using the fact that the defect theorem holds for henselian valuations (Draxl’s theorem) Theorem 2.3 is an immediate consequence of Theorem 2.4 since

[D : F] = [F^{h} ⊗_{F} D : F^{h}] = n_{V}^{2} · [D^{h} : F^{h}] = n_{V}^{2} · e_{V^{h}} · f_{V^{h}} · p^{δ} = n_{V}^{2} · e_{V} · f_{V} · p^{δ}.

3 Dubrovin valuation rings

In this section let Q be a central simple algebra with center F such that [Q : F] = n^{2} and let D be the underlying division algebra where Q ≅ M_{l}(D). Moreover, we always will assume that Q = M_{l}(D).

Definition 3.1 A subring B of Q is called Dubrovin valuation ring of Q if there exists an ideal M of B such that the following hold.

i) B/M is simple artinian. ii) For all a ∈ Q \ B there exist b, b′ ∈ B with a · b, b′ · a ∈ B \ M.

This notation of valuation ring for central simple algebras (simple artinian rings) goes back to N.I. Dubrovin (cf. [9]) and the following list of properties shows that the name valuation ring is justified (cf. [9, 10]). Let B be a Dubrovin valuation ring of Q with center V = F ∩ B. (D1) V is a valuation ring of F. (D2) The two-sided ideals of B are totally ordered by inclusion where a two-sided

ideal is a B-bimodule of Q. An immediate consequence of (D2) is that J(B) is the unique maximal ideal

of B and that M = J(B) with M as in Definition 3.1, i.e., M is also uniquely determined. Since B/M is simple, J(B) is a prime ideal of B. (D3) Each finitely generated left and each finitely generated right ideal of B is

principal as a left resp. right ideal. A subring R of Q with property (D3) is called Bezout ring or Bezout order if

R is an order in Q (here an order is a subring with Q as its total quotient ring). Since any Dubrovin valuation ring of Q is an order, property (D3) just means that any Dubrovin valuation ring is a Bezout order. (D4) Any overring B′ of B in Q is a Dubrovin valuation ring of Q and J(B′) is

a prime ideal of B. On the other hand, if P is a prime ideal of B then the classical localization B_{P} can be formed and B_{P} is a Dubrovin valuation ring of Q with J(B_{P}) = P.

The classical localization B_{P} can be formed if the set S of all s ∈ B \ P which are regular modulo P is a regular Ore set. The elements of B_{P} then can be written as bs^{−1} with b ∈ B and s ∈ S. Property (D4) also holds for left localizations.

(D5) If B′ is an overring of B in Q then B/J(B′) is a Dubrovin valuation ring of B′/J(B′).

(D6) There exists some regular x in Q and some Dubrovin valuation ring B′ of D such that xBx^{−1} = M_{l}(B′).

Examples. 1) If Q is a division algebra, i.e. Q = D, then any total valuation ring of Q is a Dubrovin valuation ring of Q.

2) If B is a Dubrovin valuation ring of D then M_{l}(B) is a Dubrovin valuation ring of M_{l}(D).

3) If V is a discrete valuation ring of F then any maximal order in Q over V is a Dubrovin valuation ring.

More specific and detailed examples can be found in [16].

The following result is central in the theory of noncommutative valuation rings.

Theorem 3.2 Any valuation ring V of F can be extended to a Dubrovin valuation ring B of Q. Furthermore, all extensions of V are conjugate.

The extension theorem was proved by N.I. Dubrovin in [10] and the conjugacy theorem by A.R. Wadsworth in [35]. Different proofs for both of these results were given later on in [4] resp. [14].

If Q is not a division algebra then the number of extensions of V to Q is infinite, since in this situation any extension of V has an infinite number of conjugates. But this is quite different if Q = D is a division algebra as the following three cases indicate.

1. V has exactly one extension to D. This happens if and only if V can be extended to an invariant valuation ring, i.e., the valuation of F belonging to V can be extended to a valuation of D.

2. V has more than one but only a finite number of extension to D. This happens if and only if V can be extended to a total valuation ring of D which is not invariant. In this case all extensions are total valuation rings which are not invariant.

3. V has an infinite number of extensions. This happens if and only if V cannot be extended to a total valuation ring.

The fact that the number of Dubrovin extensions is infinite ′′in general′′ was proved by A.R. Wadsworth in [35] and it shows that the defect theorem cannot be extended to Dubrovin valuation rings in a naive way. Before we introduce the intersection property of Dubrovin valuation rings which leads to the so-called extension number of a central valuation ring let us first recall the situation where Q is a division algebra and V can be extended to a total valuation ring (Theorem 2.2): The integral closure R of V in Q is the intersection of all extensions B_{1}, …, B_{k} of V and {J(B_{i}) ∩ R | i = 1, …, k} is the set of all maximal ideals of R. Furthermore, if B is any overring of some B_{i} then J(B) ∩ R is a prime ideal of R and any prime ideal of R can be obtained in this way. On the other hand, for any prime ideal P of R the classical localization R_{P} can be formed and R_{P} is a valuation ring of Q such that J(R_{P}) ∩ R = P. Finally, if M is a maximal ideal then R_{M} = B_{i} for some i = 1, …, k. Unfortunately, if B_{1}, …, B_{k} are Dubrovin extensions of V which are not total then their intersection does not have such a nice structure in general.

Definition 3.3 Let B_{1}, …, B_{k} be Dubrovin valuation rings of Q. Then B_{1}, …, B_{k} have the intersection property, if J(B)∩R is a prime ideal of R for any overring B of B_{i}, i = 1, …, k where R is the intersection of B_{1}, …, B_{k}.

Originally, the intersection property has been defined in terms of three conditions (cf. [14]) but in [37] it has been shown that the one used in Definition 3.3 implies the other two. This is a nice result which is far from being obvious.

Example. 1. A single Dubrovin valuation ring has the intersection property (cf. (D4)).

2. If Q = D is a division algebra then B_{1}, …, B_{k} have the intersection property whenever B_{1}, …, B_{k} are total valuation rings.

3. If B_{1}, …, B_{k} are Dubrovin valuation rings of Q having the intersection property then M_{m}(B_{1}), …, M_{m}(B_{k}) are Dubrovin valuation rings of M_{m}(Q) with intersection property.

Even though the condition used in Definition 3.3 seems to be very weak it leads to very strong ring theoretical properties as the following theorem shows (cf. [15]).

Theorem 3.4 Let B_{1}, …, B_{k} be Dubrovin valuation rings of Q. Then B_{1}, …, B_{k} have the intersection property if and only if B_{1}∩…∩B_{k} is a Bezout ring (Bezout order in Q).

If R is any Bezout order in Q then for each prime ideal P of R the classical localization R_{P} can be formed and R_{P} is a Dubrovin valuation ring of Q. Furthermore, the correspondence P ⟷ R_{P} leads to a bijection between the set of all prime ideals of R and the set of all Dubrovin valuation rings of Q containing R (cf. [15]). This shows that Bezout orders are Prüfer orders in the sense of [2], i.e., B_{1} ∩ … ∩ B_{k} is a Prüfer order if B_{1}, …, B_{k} have the intersection property. On the other hand, in [27] it is shown that the opposite conclusion also holds and this means that B_{1}, …, B_{k} have the intersection property if and only if B_{1} ∩ … ∩ B_{k} is a Prüfer order.

The intersection property is interesting also in other respects. For instance, at the same time when the intersection property was introduced, P. Morandi investigated the general approximation theorem as stated in section 2 and he proved that for Dubrovin valuation rings the approximation theorem holds if and only if a certain condition (which we will call condition (A)) is satisfied (for details cf. [25]). Since the approximation theorem was needed in [14] to prove some essential results, condition (A) and the intersection property both had to be considered, and it turned out that these two conditions are equivalent (cf. [14]). This means that the general approximation theorem holds for pairwise incomparable Dubrovin valuation rings if and only if they have the intersection property and it shows for example that B_{1} ∩ … ∩ B_{k} is a Bezout order if B_{1}, …, B_{k} are Dubrovin valuation rings with intersection property.

Theorem 3.5 Let V be a valuation ring of F. Then there exists a natural number n_{V} such that the following hold. If B_{1}, …, B_{m} are Dubrovin extensions of V with intersection property, then

1. m ≤ n_{V}.
2. m = n_{V} ⟺ B_{1} ∩ … ∩ B_{m} is integral over V.

3. If m < n_{V}, then there exist Dubrovin extensions B_{m+1}, …, B_{n_{V}} such that B_{1} ∩ … ∩ B_{n_{V}} is integral over V and B_{1}, …, B_{n_{V}} have the intersection property.

If V is a central valuation ring then first of all, there exists a Dubrovin valuation ring B_{1} of Q extending V. If B_{1} is not integral then there exists another Dubrovin extension B_{2} of V such that B_{1} ∩ B_{2} is a Bezout order, but in general not every Dubrovin extension can be choosen. Now, if B_{1} ∩ B_{2} is not integral, then this process can be continued to obtain B_{3}, B_{4}, …, until B_{1}, …, B_{k} are found such that the Bezout order B_{1} ∩ … ∩ B_{k} is integral over V. According to Theorem 3.5 this happens if and only if k = n_{V}. Thus, for any other Dubrovin extension B the intersection B_{1} ∩ … ∩ B_{n_{V}} ∩ B is not Bezout, but one might ask whether B_{1} ∩ … ∩ B_{n_{V}} ∩ B still can have nice ring theoretical properties. For instance, B_{1} ∩ … ∩ B_{n_{V}} ∩ B can be semihereditary and the reader who is interested in this topic should study [18, 19, 20] where semihereditary orders over valuation rings are investigated, especially such orders which are intersections of Dubrovin valuation rings. With respect to Theorem 3.5, the following definition is natural.

Definition 3.6 If V is a valuation ring of F then n_{V} as defined in Theorem 3.5 is called extension number of V.

Example. 1) If Q = D is a division algebra and if there exist exactly k total valuation rings of D extending V then k = n_{V}. This follows directly from the fact that total valuation rings always have the intersection property and that the intersection of all extensions of V is the integral closure of V in D whenever V has at least one total extension.

2) If B_{1}, …, B_{k} are all total valuation rings of D extending V then M_{l}(B_{1}), …, M_{l}(B_{k}) are Dubrovin extensions of V to M_{l}(D) with intersection property and M_{l}(B_{1}) ∩ … ∩ M_{l}(B_{k}) = M_{l}(B_{1} ∩ … ∩ B_{k}) is integral over V, i.e., n_{V} = k.

3) If V can be extended to an integral Dubrovin valuation ring B, i.e., B is integral over V, then n_{V} = 1. In case of a total valuation ring this happens if and only if V can be extended to an invariant valuation ring since any total valuation ring is invariant if and only if it is integral. If B is not total then there exists another extension B′ of V and B, B′ do not have the intersection property since n_{V} = 1. As a specific example we consider Q = M_{l}(F) and B = M_{l}(V) where V is a valuation ring of F. Then B is an integral extension of V and B, qBq^{−1} do not have the intersection property for any regular q ∈ Q whenever B ≠ qBq^{−1}.

In [14] the intersection property was introduced to derive the extension number where the extension number was expected to be the right one which can replace the number of all extensions in the defect theorem, since this number is infinite in general. Before we can state the defect theorem for Dubrovin valuation rings the corresponding residue degree and ramification index must be defined. But this is straightforward and similar to the definitions given in section 2. The group Γ_{B} of all invertible fractional two-sided ideals of B is an ordered abelian group which is called value group of B. The value group Γ_{V} of V can be considered as a subgroup of Γ_{B} and e_{V} = [Γ_{B} : Γ_{V}] is the ramification index of V, which is independend of the special choice of B. Furthermore, \overline{B} = B/J(B) is a central simple algebra and \overline{V} can be considered as a subfield of Z(\overline{B}) which is a normal field extension of \overline{V} such that Gal(Z(\overline{B})/\overline{V}) is a homomorphic image of Γ_{B}/Γ_{V} (cf. [35]). We call f_{V} := [\overline{B} : \overline{V}] the residue degree of V and f_{V} does not depend on the special choice of B.

Theorem 3.7 Let V be a valuation ring of F with extension number n_{V}. Then

[Q : F] = e_{V} · f_{V} · n_{V}^{2} · p^{δ}

where p is the characteristic of \overline{V} and δ a nonnegative integer.

The proof of Theorem 3.7 is rather intricate and two different approaches are known. In [14] a proof is given which is sketched in section 2 for total valuation rings and where, for instance, invariant valuation rings have to be replaced by integral Dubrovin valuation rings and the number of extensions of a central valuation ring by the extension number. The main arguments are the same but the details are more complicated. In [35] a different version of the defect theorem is proved, which is the first one stated for Dubrovin valuation rings. In that paper A.R. Wadsworth investigates what happens to a Dubrovin valuation ring with passage to the Henselization of its center. The following theorem only reflects some of the results given in [35].

Theorem 3.8 Let V be a valuation ring of F and let B be a Dubrovin valuation ring of Q extending V. Furthermore, let \overline{B} ≅ M_{t_{B}}(E) where E is the underlying division algebra of \overline{B} and t_{B} its matrix size and let

F^{h} ⊗_{F} Q ≅ M_{n_{B}}(D^{h})

where D^{h} is the underlying division algebra of F^{h} ⊗_{F} Q and n_{B} its matrix size. If B^{h} denotes the unique extension of V^{h} to D^{h} then the following hold.

1. \overline{B} ≅ M_{t_{B}}(\overline{B^{h}}).

2. [Γ_{B^{h}} : Γ_{V^{h}}] = e_{V}.

3. t_{B} is a divisor of n_{B}.

To derive Wadsworth’s version of the defect theorem we use Draxl’s theorem and obtain

[Q : F] = [F^{h} ⊗_{F} Q : F^{h}]
= n_{n_{B}}^{2} · [D^{h} : F^{h}]
= n_{B}^{2} · e_{V^{h}} · f_{V^{h}} · p^{δ}
= (\frac{n_{B}}{t_{B}})^{2} · e_{V} · (t_{B}^{2} · f_{V^{h}}) · p^{δ}
= (\frac{n_{B}}{t_{B}})^{2} · e_{V} · f_{V} · p^{δ},

where \frac{n_{B}}{t_{B}} is a natural number. Comparing this result and Theorem 3.7 one would expect that \frac{n_{B}}{t_{B}} is the extension number of V, and indeed this is proved in [14].

Theorem 3.5 and Theorem 3.4 show that any central valuation ring V can be extended to a Bezout order R of Q which is integral over V and which is the intersection of n_{V} Dubrovin extensions of V. The following theorem shows that any integral Bezout order extending V can be obtained in this way and that these rings can be considered as ′′almost integral closure′′ of V in Q (cf. [15]).

Theorem 3.9 Let V be a valuation ring of F.

1. There exists a Bezout order R of Q extending V (which is integral over V).

2. A Bezout order R of Q extending V is integral over V if and only if R is minimal with respect to the inclusion.

3. If R is a Bezout order of Q extending V then R = R_{M_{1}} ∩ … ∩ R_{M_{k}} where {M_{1}, …, M_{k}} is the set of all maximal ideals of R and any R_{M_{i}} is a Dubrovin valuation ring of Q extending V,i.e., R_{M_{1}}, …, R_{M_{k}} have the intersection property.

4. Any two integral Bezout orders R, R′ extending V are conjugate, i.e., there exists a regular q ∈ Q such that R′ = qRq^{−1}.

5. ⋃ qRq^{−1} is the integral closure of V in Q where the union runs over all regular q ∈ Q.

A ring B is a total valuation ring of a division algebra D if and only if B is a Bezout order in D and B/J(B) is a division algebra. Similarly, a subring B of a simple artinian ring Q is a Dubrovin valuation ring if and only if B is a Bezout order in Q with B/J(B) simple artinian. Therefore, it seems natural to call a subring B in a semisimple artinian ring Q a valuation ring if B is a Bezout order in Q and B/J(B) is semisimple artinian. This class of valuation rings for semisimple artinian rings is introduced in [5], where it is also proved that many properties of Dubrovin valuation rings carry over to the semisimple case, especially if B is in addition a ring with polynomial identity.

References

[1] Azumaya, G. On maximally central algebras. Nagoya Math. J. 2 (1951), 119-150.

[2] Alajbegović, J. H., Dubrovin, N. I. Noncommutative Prüfer rings. J. Algebra 135 (1990), 165-176.

[3] Brungs, H. H., Gräter, J. Valuation rings in finite-dimensional division algebras. J. Algebra 120 (1989), 90-99 .

[4] Brungs, H. H., Gräter, J. Extensions of valuation rings in central simple algebras. Trans. Amer. Math. Soc. 317 (1990), 287-302.

[5] Brungs, H. H., Gräter, J. Orders of higher rank in semisimple Artinian rings. Math. Nachr. 182 (1996), 73-88.

[6] Brungs, H. H., Gräter, J. Trees and valuation rings. Trans. Amer. Math. Soc. (to appear).

[7] Cohn, P. M. On extending valuations in division algebras. Studia Sci. Math. Hungar. 16 (1981), 65-70.

[8] Draxl, P. Ostrowski’s theorem for Henselian valued skew fields. J. Reine Angew. Math. 354 (1984), 213-218.

[9] Dubrovin, N. I. Noncommutative valuation rings. Trudy Moskov. Mat. Obsch. 45 (1982), 265-280; English transl., Trans. Moscow Math. Soc. 45 (1984), 273-287.

[10] Dubrovin, N. I. Noncommutative valuation rings in simple finite-dimensional algebras over a field. Mat. Sb. 123 (1984), 496-509; English transl., Math. USSR-Sb. 51 (1985), 493-505.

[11] Dubrovin, N. I. Noncommutative Prüfer rings. Mat. USSR, Sb. 74 (1993), 1-8; English transl., Math. Sb. 182 (1991), 1251-1260.

[12] Endler, O. Valuation Theory. Springer, 1972.

[13] Gräter, J. Lokalinvariante Bewertungen. Math. Z. 192 (1986), 183-194.

[14] Gräter, J. The Defektsatz for central simple algebras. Trans. Amer. Math. Soc. 330 (1992), 823-843.

[15] Gräter, J. Prime PI-rings in which finitely generated right ideals are principal. Forum Math. 4 (1992), 447-463.

[16] Haile, D. E., Morandi, P. J. On Dubrovin valuation rings in crossed product algebras. Trans. Amer. Math. Soc. 338 (1993), 723-751.

[17] Haile, D. E., Morandi, P. J., Wadsworth, A. R. Bezout orders and Henselization. J. Algebra 173 (1995), 394-423.

[18] Kauta, J. S. Integral semihereditary orders, extremality, and Henselization. J. Algebra 189 (1997), 226-252.

[19] Kauta, J. S. Integral semihereditary orders inside a Bezout maximal order. J. Algebra 189 (1997), 253-272.

[20] Kauta, J. S. On semihereditary maximal orders. Bull. London Math. Soc. 30 (1998), 251-257.

[21] Krull, W. Allgemeine Bewertungstheorie. J. Reine Angew. Math. 167 (1932), 160-196.

[22] Marubayashi, H., Miyamoto, H., Ueda, A. Non-commutative Valuation Rings and Semihereditary Orders. K-Monographs in Mathematics vol. 3, Kluwer, 1997.

[23] Mathiak, K. Valuations of Skew Fields and Projective Hjelmslev Spaces. LNM vol. 1175, Springer, 1986.

[24] Morandi, P. J. The Henselization of a valued division algebra. J. Algebra 122 (1989), 232-243.

[25] Morandi, P. J. An approximation theorem for Dubrovin valuation rings. Math. Z. 207 (1991), 71-82.

[26] Morandi, P. J. Maximal orders over valuation rings. J. Algebra 152 (1992), 313-341.

[27] Morandi, P. J. Noncommutative Prüfer rings satisfying a polynomial identity. J. Algebra 161 (1993), 324-341.

[28] Morandi, P. J., Wadsworth, A. R. Integral Dubrovin valuation rings. Trans. Amer. Math. Soc. 315 (1989), 623-640.

[29] Pierce, R. S. Associative Algebras. Grad. Texts in Math. vol. 88, Springer, 1982.

[30] Reiner, I. Maximal Orders. Academic Press, London, 1975.

[31] Schilling, O. F. G. Noncommutative Valuations. Bull. Amer. Math. Soc. 51 (1945), 297-304.

[32] Schilling, O. F. G. The Theory of Valuations. Math. Surveys and Monographs vol. 4, Amer. Math. Soc. Providence, 1950.

[33] Wadsworth, A. R. Extending valuations to finite-dimensional division algebras. Proc. Amer. Math. Soc. 98 (1986), 20-22.

[34] Wadsworth, A. R. Dubrovin valuation rings. Perspectives in ring theory (F. van Oystaeyen and L. LeBruyn, eds.), NATO ASI Series, Series C 233, Kluwer, Dordrecht, 1986, 359-374.

[35] Wadsworth, A. R. Dubrovin valuation rings and Henselization. Math. Ann. 283 (1989), 301-328.

[36] Wadsworth, A. R. Valuations on Tensor Products of Symbol Algebras. Contemporary Math. 124 (1992), 275-289.

[37] Zhao, Y. On the intersection property of Dubrovin valuation rings. Proc. Amer. Math. Soc. 125 (1997), 2825-2830.