## Non-commutative valuation rings of $$K(X;\sigma,\delta)$$ over a division ring $$K$$.(English)Zbl 1066.16051

Let $$K$$ be a skew field with an endomorphism $$\sigma$$ and a $$\sigma$$-derivation $$\delta$$. The authors consider the skew polynomial ring $$A=K[X;\sigma,\delta]$$ in an indeterminate $$X$$ [see; e.g.; P. M. Cohn, Further Algebra and Applications, London: Springer (2003; Zbl 1006.00001), 7.3] and its skew field of fractions $$F=K(X;\sigma,\delta)$$ and study the valuation rings in $$F$$ and their relation to $$A$$.
The $$\sigma$$-derivation $$\delta$$ is called quasi-algebraic if there is a polynomial in $$\delta$$ of positive degree $$n$$ over $$K$$ which is the inner $$\sigma^n$$-derivation induced by its constant term. They observe that if $$\delta$$ is not quasi-algebraic, then there are no proper (i.e., $$\neq F$$) non-commutative valuation rings of $$F$$ containing $$A$$. Thus to determine the valuation rings containing $$A$$ they may assume $$\delta$$ to be quasi-algebraic.
First they describe the maximal ideals of $$A$$, following C. Cauchon [Thèse, Orsay (1977)]. Let $$p(X)$$ be the monic invariant polynomial of least degree $$n>0$$; if the inner order of $$\sigma$$ is infinite (i.e., $$\sigma^r$$ is never an inner automorphism for $$r\neq 0)$$ then the only maximal ideal is $$pA= Ap$$; otherwise there are also others, described by irreducible polynomials in powers of $$p(x)$$ over the centre of $$K$$. A prime Goldie ring in a simple Artinian ring is called a Dubrovin valuation ring if it is local and semi-hereditary. Suppose now that $$\sigma$$ is an automorphism of $$K$$ and $$\sigma$$ a quasi-algebraic $$\sigma$$-derivation with $$p(x)$$ a monic invariant polynomial of least positive degree. Then the authors show i) if $$\sigma$$ has infinite inner order then the only proper Dubrovin valuation ring of $$F$$ containing $$A$$ is $$A_P$$, where $$P=Ap(x)$$; ii) if $$\sigma$$ has inner order $$m$$ then the proper Dubrovin valuation rings containing $$A$$ are the localizations of $$A$$ by maximal ideals as described above.
If $$\sigma$$ is not an endomorphism, then $$pKp^{-1}$$ is a proper subfield (if coefficients are written on the left), so that $$K\subset K_1\subset K_2\subset\dots$$, $$K_i=p^{-i}Kp^i$$ is an ascending chain; write its union as $$\overline K$$ and define $$\overline A$$ similarly. Then $$\sigma$$ induces an automorphism $$\overline\sigma$$ of $$\overline K$$ which is of infinite inner order. The authors now show that $$\overline A=\overline K[X;\overline\sigma,\overline\delta]$$ for a $$\overline\sigma$$-derivation $$\overline\delta$$ and if $$\sigma$$ is quasi-algebraic, then $$\overline Ap(X)$$ is the unique maximal ideal of $$\overline A$$. For the case where $$\overline K$$ is algebraic over $$K$$, they are able to describe the total valuation rings of $$F$$ contained in $$\overline A_{\overline P}$$ in certain cases. This situation is illustrated by some examples.
Secondly the authors consider $$K,\sigma,\delta$$ as before and a total valuation ring $$V$$ of $$K$$, such that both $$V$$ and its maximal ideal $$J(V)$$ admit $$\sigma$$ and $$\delta$$. Let $$A^{(1)}$$ be the localization of $$V[X;\sigma,\delta]$$ at the ideal generated by $$J(V)$$. This is a total valuation ring of $$F$$ such that $$A^{(1)}\cap K=V$$ and they make a study of valuation rings $$B$$ in $$F$$ such that $$B\cap K=V$$, $$B\subseteq A^{(1)}$$ and $$X\in B$$. A left order $$S$$ in a simple Artinian ring $$Q$$ is called left Prüfer if any finitely generated left $$S$$-ideal in $$Q$$ is a progenerator of the category of left $$S$$-modules. The authors show that if $$S$$ is a Dubrovin valuation ring of a simple Artinian ring $$Q$$ and $$f\colon S\to\overline S=S/J(S)$$ is the natural homomorphism, while $$R$$ is a left order in $$\overline S$$ with complete inverse image $$A$$ in $$S$$, then 1) $$A$$ is a left order in $$Q$$, 2) $$A$$ is left Prüfer in $$Q$$ if and only if $$R$$ is left Prüfer in $$\overline S$$, 3) If $$A$$ is left Prüfer and $$\mathfrak p$$ is a prime ideal in $$R$$ with inverse image $$\mathfrak P$$ a prime ideal in $$A$$, then $$\mathfrak p$$ is left localizable if and only if $$\mathfrak P$$ is left localizable and then $$A_{\mathfrak p}=f^{-1}(R_{\mathfrak P})$$, 4) $$A$$ is a Dubrovin valuation ring if and only if $$R$$ is one, 5) If $$Q$$ is a skew field with total valuation subring $$S$$, then $$A$$ is a total valuation ring if and only if $$R$$ is one.
As examples show, the invariance of a valuation subring $$A$$ of a skew field $$Q$$ is not equivalent to $$R$$ and $$S$$ being invariant (under inner automorphisms). Further examples show the different situations that can arise.

### MSC:

 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16S36 Ordinary and skew polynomial rings and semigroup rings 16D25 Ideals in associative algebras

Zbl 1006.00001
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