Non-commutative valuation rings of \(K(X;\sigma,\delta)\) over a division ring \(K\).
type: Item
instance of: scholarly article
title: Non-commutative valuation rings of \(K(X;\sigma,\delta)\) over a division ring \(K\).
zbMATH Open document ID: 1066.16051
author: H. Marubayashi
author: Shigeru Kobayashi
author: H. Komatsu
author: Guangming Xie
published in: Journal of the Mathematical Society of Japan
publication date: 2004-10-28T00:00:00Z
review text: 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.; \textit{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 \textit{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 \(\sigma\)…
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 2111273
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: skew fields of fractions
zbMATH Keywords: Dubrovin valuation rings
zbMATH Keywords: maximal ideals
zbMATH Keywords: automorphisms
zbMATH Keywords: derivations
zbMATH Keywords: localizations
zbMATH Keywords: total valuation rings
zbMATH Keywords: left orders
zbMATH Keywords: prime ideals
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191334084
OpenAlex ID: W2005679113
DOI: 10.2969/JMSJ/1191334084
description: scientific article; zbMATH DE number 2111273