NON-COMMUTATIVE VALUATION RINGS OF K(X; σ, δ) OVER A DIVISION RING K

Shigeru KOBAYASHI  Hiroaki KOMATSU
Hidetoshi MARUBAYASHI  Guangming XIE ^{∗}

Abstract Let K be a division ring with a σ-derivation δ, where σ is an endomorphism of K and K(X; σ, δ) be the quotient division ring of the Ore extension K[X; σ, δ] over K in an indeterminate X. Suppose that (σ, δ) is compatible with V, where V is a total valuation ring of K, then R^{(1)} = V [X; σ, δ]_{J(V)[X;σ,δ]}, the localization of V [X; σ, δ] at J(V)[X; σ, δ], is a total valuation ring of K(X; σ, δ). The aim of this paper is to describe non-commutative valuation rings B of K(X; σ, δ) such that R^{(1)} ⊇ B, B ∩ K = V and X ∈ B.

1. Introduction

Let K be a division ring, σ be an endomorphism of K and δ be a σderivation, i.e., an additive map δ : K ⟶ K such that δ(ab) = σ(a)δ(b)+δ(a)b for all a, b ∈ K. As usaual, K[X; σ, δ] = {f(X) | f(X) = a_{n}X^{n} + ⋯ + a_{0}, a_{i} ∈ K} is the Ore extension over K with Xa = σ(a)X + δ(a) for any a ∈ K, where X is an indeterminate. Let V be a total valuation ring of K and (σ, δ) be compatible with V. Then, in [BT], they proved by Krull methed that J(V)[X; σ, δ] is a left Ore set of V [X; σ, δ] and R^{(1)} = V [X; σ, δ]_{J(V)[X;σ,δ]}, the localization of V [X; σ, δ] at J(V)[X; σ, δ], is a total valuation ring of K(X; σ, δ) such that R^{(1)} ∩ K = V, X ∈ R^{(1)}. The aim of this paper is to describe noncommutative valuation rings B of K(X; σ, δ) such that R^{(1)} ⊇ B, B ∩ K = V and X ∈ B.

There are three types of non-commutative valuation rings as follows: Let Q be a simple Artinian ring and let R be an order in Q, i.e., R is a prime Goldie ring. We say that R is a Dubrovin valuation ring of Q if R is semihereditary and R is local, i.e., R/J(R) is a simple Artinian ring, where J(R) is the Jacobson radical of R. Assume that Q is a division ring. A subring R of Q is said to be a total valuation ring of Q, if for any non-zero q ∈ Q, either q ∈ R or q^{−1} ∈ R. In the case where K is a field, the definition of total valuation rings is the same as one of the equivalent definitions of [G, (16.3)]. However, in the case where K is not a field, the structure of total valuation rings is much different from one of commutative valuation rings (see, [D_{2}], [BMO]). Furthermore, a total valuation ring R of Q is said to be invariant if q^{−1}Rq = R for all non-zero q ∈ Q. It is easy to see that a total valuation ring R is a Dubrovin valuation ring and the converse does not necessarily hold. We refer to [MMU] for more properties of non-commutative valuation rings.

∗The detailed version of this paper has been submitted for publication elsewhere

2. Non-commutative valuation rings of K(X; σ, δ) contained in R^{(1)}

Let K be a division ring, σ be an endomorphism of K, δ be a σ-derivation and V be a total valuation ring of K. We assume that (σ, δ) is compatible with V, i.e., σ(V) ⊆ V, σ(J(V)) ⊆ J(V), δ(V) ⊆ V, δ(J(V)) ⊆ J(V). In [BT], they proved that J(V)[X; σ, δ] is left localizable and R^{(1)} = V [X; σ, δ]_{J(V)[X;σ,δ]}, the localization of V [X; σ, δ] at J(V)[X; σ, δ], is a total valuation ring of K(X; σ, δ) with R^{(1)} ∩ K = V, X ∈ R^{(1)} and we studied some properties of R^{(1)} ( see [XKM]). In this paper we shall study non-commutative valuation rings B of K(X; σ, δ) such that B ∩ K = V, B ⊆ R^{(1)} and X ∈ B. For simplicity, we denote by D the set of all Dubrovin valuation rings B of K(X; σ, δ) such that B ∩ K = V, R^{(1)} ⊇ B and X ∈ B, by T the set of all total valuation rings B of K(X; σ, δ) such that B ∩ K = V, R^{(1)} ⊇ B and X ∈ B. Obviously, T ⊆ D. To get some information of D and T , we use the following natural map.

Let φ : R^{(1)} = V [X; σ, δ]_{J(V [X;σ,δ])} ⟶ \overline{R^{(1)}} = R^{(1)}/J(R^{(1)}) ≅ \overline{V} (X; \overline{σ}, \overline{δ}) be the natural homomorphism, where \overline{σ}(\overline{v}) = [σ(v) + J(V)] and \overline{δ}(\overline{v}) = [δ(v) + J(V)] for any \overline{v} = [v + J(V)] ∈ \overline{V} = V/J(V). Set R = φ^{−1}(\overline{V} [X; \overline{σ}, \overline{δ}]) = V [X; σ, δ] + J(R^{(1)}).

The following lemma is crucial to find out non-commutative valuation rings in D.

Lemma 2.1 There is a one-to-one correspondence between D and the set of all Dubrovin valuation rings 𝔅 of \overline{V} (X; \overline{σ}, \overline{δ}) with 𝔅 ⊇ \overline{V} [X; \overline{σ}, \overline{δ}], which is given by φ(B) = 𝔅 and φ^{−1}(𝔅) = B, where B ∈ D.

Proof Let B ∈ D with B ≠ R^{(1)}. Then B ⊋ J(R^{(1)}) and φ(B) = B/J(R^{(1)}) is a Dubrovin valuation ring of \overline{R^{(1)}} (see [MMU, (6.6)]).

Conversely, it is not difficult to prove that for any Dubrovin valuation ring 𝔅 of \overline{V} (X; \overline{σ}, \overline{δ}) with 𝔅 ⊇ \overline{V} [X; \overline{σ}, \overline{δ}], B = φ^{−1}(𝔅) is in D.

Remark: Since R^{(1)} is a total valuation ring, we can replace ”D” and ”Dubrovin valuation rings” by ”T ” and ”total valuation rings” respectively in Lemma 2.1.

We know from Lemma 2.1 that it suffices to study Dubrovin valuation rings 𝔅 of \overline{V} (X; \overline{σ}, \overline{δ}) with 𝔅 ⊇ \overline{V} [X; \overline{σ}, \overline{δ}] in order to describe Dubrovin valuation rings in D. Let 𝔅 be any proper Dubrovin valuation ring of \overline{V} (X; \overline{σ}, \overline{δ}) with 𝔅 ⊇ \overline{V} [X; \overline{σ}, \overline{δ}]. Then J(𝔅)∩\overline{V} [X; \overline{σ}, \overline{δ}] is a non-zero ideal of \overline{V} [X; \overline{σ}, \overline{δ}]. Since \overline{V} [X; \overline{σ}, \overline{δ}] is a left principal ideal domain (note that \overline{V} is a division ring), some 𝔅 are obtained by the localization of \overline{V} [X; \overline{σ}, \overline{δ}] at some maximal ideals. These show that there are close relatons between the ideal theory of \overline{V} [X; \overline{σ}, \overline{δ}] and non-commutative valuation rings of \overline{V} (X; \overline{σ}, \overline{δ}). So we will firstly study the ideal theory of K[X; σ, δ], where K is a division ring, which is deeply depend on the properties of σ and δ. In order to study the ideal theory of K[X; σ, δ], we will introduce the following which was defined in [L] (also see [LL]). δ is said to be a quasi-algebraic σ-derivation if there exist a_{n} = 1, a_{n−1}, ⋯, a_{0} ∈ K, n > 0 such that Σ_{i=1}^{n}a_{i}δ^{i} = D_{a_{0},σ^{n}}, where D_{a_{0},σ^{n}}(b) = a_{0}b − σ^{n}(b)a_{0} for any b ∈ K. The inner order of σ, denoted by ∘(σ), is defined by the smallest positive integer n such that σ^{n} = I_{a}, the inner automorphism induced by some a ∈ K; if no such natural number n exists, then ∘(σ) is ∞. A monic polynomial p(X) with deg p(X) = n is said to be invariant if for any a ∈ K, p(X)a = σ^{n}(a)p(X), p(X)X = (X + c)p(X) for some c ∈ K.

The following lemma was proved by Lam, Leung, Leroy and Matczuk (see [LLLM, 3.6]) after long history starting from Amitsur [A], in the case where σ = 1.

Lemma 2.2 K[X; σ, δ] is simple if and only if δ is not a quasi-algebraic σ-derivation.

In the case where δ is a quasi-algebraic σ-derivation, there exists a monic invariant polynomial p(X) of minimal non-zero degree from Lemma 2.2 which plays a very important role to study the ideal theory of K[X; σ, δ] as follows.

Lemma 2.3 (Cauchon) Let δ be a quasi-algebraic σ-derivation. Then there is a monic invariant polynomial p(X) of minimal non-zero degree.

(1) If ∘(σ) = ∞, then Z(K[X; σ, δ]) = Z(K)_{σ,δ}, where Z(S) stands for the center of S for any ring S and Z(K)_{σ,δ} = {a ∈ Z(K) | σ(a) = a and δ(a) = 0}.

(2) If ∘(σ) = n < ∞, then Z(K[X; σ, δ]) = Z(K)_{σ,δ}[λp(X)^{l}] for some non-zero λ ∈ K and some natural number l.

Lemma 2.4 Let δ be a quasi-algebraic σ-derivation and p(X) be a monic invariant polynomial of minimal non-zero degree.

(1) If ∘(σ) = ∞, then P = K[X; σ, δ]p(X) is the only maximal ideal of K[X; σ, δ].

(2) If ∘(σ) = n < ∞, then any maximal ideal of K[X; σ, δ] is one of the following: P = K[X; σ, δ]p(X), M = K[X; σ, δ]w(Y), where Y = λp(X)^{l} for some λ ∈ K and w(Y) runs over all irreducible polynomials of Z(K)_{σ,δ}[Y] different from Y.

We will study non-commutative valuation rings in D in the following five cases.

Theorem 2.5 (case 1) If \overline{δ} is not a quasi-algebraic \overline{σ}-derivation, then D =\{R^{(1)}\}.

Proof Let B ∈ D with B ≠ R^{(1)}. Then 𝔅 = φ(B) is a proper Dubrovin valuation ring of \overline{V} (X; \overline{σ}, \overline{δ}) containing \overline{V} [X; \overline{σ}, \overline{δ}] and so J(𝔅) ∩ \overline{V} [X; \overline{σ}, \overline{δ}] is a non-zero ideal of \overline{V} [X; \overline{σ}, \overline{δ}] which is a contradiction to Lemma 2.2.

From now on we assume that \overline{δ} is a quasi-algebraic \overline{σ}-derivation. Then we can find a monic polynomial p(X) ∈ V [X; σ, δ] such that φ(p(X)) = \overline{p(X)} ∈ \overline{V} [X; \overline{σ}, \overline{δ}] is a monic invariant polynomial of mininal non-zero degree. Let P = Rp(X). At first we study the case \overline{σ} ∈ Aut(\overline{V}). In this case \overline{V} [X; \overline{σ}, \overline{δ}] is a Dedekind order in \overline{V} (X; \overline{σ}, \overline{δ}). Then we can use the following lemma which follows from [D_{1}, Theorems 3 and 4, §2].

Lemma 2.6 Let Q be a simple Artinian ring and S be a Dedekind order in Q. Then there is a one-to-one correspondence between the set of all proper Dubrovin valuation rings of Q containing S and the set of all non-zero maximal ideals of S, which is given by φ : B ⟶ J(B) ∩ S and φ^{−1} : P ⟶ S_{P}, where B is a proper Dubrovin valuation ring of Q containing S and P is a non-zero maximal ideal of S.

Theorem 2.7 Suppose that \overline{δ} is a quasi-algebraic \overline{σ}-derivation and \overline{σ} ∈ Aut(\overline{V}).

(1) (case 2) If ∘(\overline{σ}) = ∞, then D =\{R^{(1)}, R_{P} \}, where P = Rp(X).

(2) (case 3) If ∘(\overline{σ}) = m < ∞, then D= \{R^{(1)}, R_{P} \} ∪ \{R_{M} | M = Rw(X), where w(X) ∈ V [X; σ, δ] such that \overline{w(X)} is an irreducible polynomial of Z(\overline{V})_{\overline{σ},\overline{δ}}[Y](Y = \overline{λ}\overline{p(X)}^{l} for some \overline{λ} ∈ \overline{V} and l ≥ 1 as in Lemma 2.4 \}. In particular, J(R_{P}) = R_{P} p(X) = p(X)R_{P} and J(R_{M}) = R_{M}w(X) = w(X)R_{M}.

Proof Since p(X), w(X) ∈ U(R^{(1)}), we easily have P = φ^{−1}(\overline{V} [X; \overline{σ}, \overline{δ}]\overline{p(X)}) = Rp(X) = p(X)R and M = φ^{−1}(\overline{V} [X; \overline{σ}, \overline{δ}]\overline{w(X)}) = Rw(X) = w(X)R, where R = V [X; σ, δ] + J(R^{(1)}). Hence the theorem follows from Lemmas 2.3, 2.4 and 2.6.

Now we consider the case where \overline{δ} is a quasi-algebraic \overline{σ}-derivation and \overline{σ} ∉ Aut(\overline{V}). Let \overline{p(X)} be a monic invariant polynomial of \overline{V} [X; \overline{σ}, \overline{δ}] with minimal non-zero degree. Then \overline{p(X)}\overline{V} [X; \overline{σ}, \overline{δ}] ⊊ \overline{V} [X; \overline{σ}, \overline{δ}]\overline{p(X)}, we can get two ascending chain:

\overline{V} [X; \overline{σ}, \overline{δ}] ⊊ \overline{p(X)}^{−1}\overline{V} [X; \overline{σ}, \overline{δ}]\overline{p(X)} ⊊ \overline{p(X)}^{−2}\overline{V} [X; \overline{σ}, \overline{δ}]\overline{p(X)}^{2} ⊊ ⋯

\overline{V} ⊊ \overline{p(X)}^{−1}\overline{V} \overline{p(X)} ⊊ \overline{p(X)}^{−2}\overline{V} \overline{p(X)}^{2} ⊊ ⋯

Let ˆ \overline{V} = ∪_{m=1}^{∞}\overline{p(X)}^{−m}\overline{V} \overline{p(X)}^{m}, a division ring containing \overline{V}, ˆ \overline{R} = ∪_{m=1}^{∞}\overline{p(X)}^{−m}\overline{R}\overline{p(X)}^{m}, where \overline{R} = \overline{V} [X; \overline{σ}, \overline{δ}]. For any α = \overline{p(X)}^{−l}\overline{a}\overline{p(X)}^{l} ∈ ˆ \overline{V}, define ˆ \overline{σ}(α) = \overline{p(X)}^{−l}\overline{σ}(\overline{a})\overline{p(X)}^{l}. We can check that ˆ \overline{σ} is well-defined and an automorphism of ˆ \overline{V} with ∘(ˆ\overline{σ}) = ∞.

We define ˆ \overline{δ} as follows; let α = \overline{p(X)}^{−l}\overline{a}\overline{p(X)}^{l} be any element of ˆ \overline{V}, where \overline{a} ∈ \overline{V}. Since

\overline{p(X)}^{l}X = (X + \overline{b_{l}})\overline{p(X)}^{l} for some \overline{b_{l}} ∈ \overline{V}, we have (X + \overline{b_{l}})\overline{a} = \overline{σ}(\overline{a})X + \overline{δ}(\overline{a}) + \overline{b_{l}}\overline{a} = \overline{σ}(\overline{a})(X + \overline{b_{l}}) + \overline{a_{l}},

where \overline{a_{l}} = \overline{δ}(\overline{a}) + \overline{b_{l}}\overline{a} − \overline{σ}(\overline{a})\overline{b_{l}} ∈ \overline{V}.

Define ˆ \overline{δ}(α) = \overline{p(X)}^{−l}\overline{a_{l}}\overline{p(X)}^{l} ∈ ˆ \overline{V}. Then

\overline{p(X)}^{l}Xα = \overline{p(X)}^{l}X\overline{p(X)}^{−l}\overline{a}\overline{p(X)}^{l}

= (X + \overline{b_{l}})\overline{a}\overline{p(X)}^{l}
= (\overline{σ}(\overline{a})(X + \overline{b_{l}}) + \overline{a_{l}})\overline{p(X)}^{l}. So
Xα = \overline{p(X)}^{−l}(\overline{σ}(a)(X + \overline{b_{l}}) + \overline{a_{l}})\overline{p(X)}^{l}
= \overline{p(X)}^{−l}\overline{σ}(\overline{a})\overline{p(X)}^{l}X + \overline{p(X)}^{−l}\overline{a_{l}}\overline{p(X)}^{l} = ˆ \overline{σ}(\overline{α})X + ˆ \overline{δ}(α).

This means ˆ \overline{δ}(α) = Xα − ˆ \overline{σ}(\overline{α})X. So ˆ \overline{δ} is well-defined. Furthermore, we can easily check that ˆ \overline{δ} is a ˆ \overline{σ}-derivation and ˆ \overline{R} = ˆ \overline{V} [X; ˆ \overline{σ}, ˆ \overline{δ}]. Let R̂ = ∪_{m=1}^{∞}p(X)^{−m}Rp(X)^{m}, an over ring of R. It is easy to see that R̂ = φ^{−1}( ˆ\overline{R}), an Prüfer order in K(X; σ, δ). Let ˆ \overline{P} = ˆ \overline{R}\overline{p(X)}. Then ˆ \overline{P} is the unique maximal ideal of ˆ \overline{R}. Let P̂ = φ^{−1}( ˆ\overline{P}). We can get R̂P̂ ∈ D.

We say that ˆ \overline{V} is algebraic over \overline{V} if for any \overline{α} ∈ ˆ \overline{V}, there exist \overline{c_{s}}, ⋯, \overline{c_{0}} ∈ \overline{V}, not all zero, such that Σ_{i=0}^{s}c̄_{i}ᾱ^{i} = 0. It is not difficult to prove the following lemma.

Lemma 2.8 If ˆ \overline{V} is algebraic over \overline{V}, then any total valuation ring B of \overline{V} (X; \overline{σ}, \overline{δ}) with \overline{B} ⊇ \overline{V} [X; \overline{σ}, \overline{δ}] containing ˆ \overline{V} [X; ˆ \overline{σ}, ˆ \overline{δ}].

If \overline{P} is a completely prime ideal of \overline{V} [X; \overline{σ}, \overline{δ}], then it is easy to prove that ˆ\overline{P} is a completely prime ideal of ˆ \overline{V} [X; ˆ \overline{σ}, ˆ \overline{δ}]. In this case R̂P̂ is a total valuation ring of K(X; σ, δ). Now by Lemmas 2.1, 2.3,2.6, and 2.8, we can get that T = {R^{(1)}, R̂P̂}. Thus we get the following theorem.

Theorem 2.9 (case 4) Suppose that \overline{δ} is a quasi-algebraic \overline{σ}-derivation and \overline{σ} ∉ Aut(\overline{V}). Then

(1) R̂P̂ ∈ D.

(2) If ˆ \overline{V} is algebraic over \overline{V} and ˆ \overline{P} is completely prime, then T = \{R^{(1)}, R̂P̂\}.

If ˆ \overline{V} is not algebraic over \overline{V}, then it becomes very sophisticated, even in the case where \overline{V} is a field. In this case, ˆ \overline{P} is a completely prime ideal of \overline{V} [X; \overline{σ}, \overline{δ}] and R̂P̂ is a total valuation ring. By using some properties of commutative valuation rings in polynomial rings and localizations, we can construct total valuation rings {A_{i}}, {B_{i}} satisfying the following properties.

Theorem 2.10 (case 5) Suppose that \overline{δ} is a quasi-algebraic \overline{σ}-derivation and \overline{σ} ∉ Aut(\overline{V}). If ˆ \overline{V} is not algebraic over \overline{V} and \overline{V} is a field with α = tr.deg\overline{V} ˆ \overline{V}, then α is infinite and there are {A_{i}}, {B_{i}} ⊆ D satisfying;

(i) R̂P̂ ⊋ A_{i} for any i ∈ Λ and A_{i} are incomparable each other, where Λ is an index set with |Λ| = α.

(ii) B_{1} ⊊ B_{2} ⊊ ⋯ ⊊ B_{α} ⊊ R̂P̂.

We have studied Dubrovin valuation rings in D classifying them into five cases. We can provide examples of total valuation rings satisfying the properties in each case. However, we will not mention the examples because of the limited pages. We refer the readers to the detailed version of this paper for examples.

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Department of Mathematics, Naruto University of Education Takashima, Naruto, 772-8502, Japan E-mail address: {skoba, marubaya, e0220104}@naruto-u.ac.jp

Faculty of Computer Science and System Engineering Okayama Prefectural University Kuboki, Soja, Okayama, Japan E-mail address: komatsu@pine.cse.oka-pu.ac.jp