##
**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.

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.

Reviewer: Paul M. Cohn (London)

### 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 |