Ore extensions over total valuation rings.
type: Item
instance of: scholarly article
title: Ore extensions over total valuation rings.
zbMATH Open document ID: 1213.16017
author: H. Marubayashi
published in: Algebras and Representation Theory
publication date: 2010-11-23T00:00:00Z
review text: Let \(K\) be a division ring, \(K^*\) its multiplicative group, and \(V\) a total valuation ring of \(K\), i.e. a subring of \(K\), such that \(k\in V\) or \(k^{-1}\in V\), for each \(k\in K^*\). Also, let \(J(V)\) be the Jacobson radical of \(V\), \(\sigma\) an automorphism of \(V\), and \(\delta\) a (left) \(\sigma\)-derivation of \(V\). We say that \((\sigma,\delta)\) is compatible with \(V\), if \(\delta(J(V))\subseteq J(V)\). The Ore extension \(R=V[x;\sigma,\delta]\) is defined as the polynomial ring over \(V\) in an indeterminate \(x\) with the multiplication defined by the rule \(xa=\sigma(a)x+\delta(a)\), for each \(a\in V\). The quotient ring of the Ore extension \(K[x;\sigma,\delta]\) of \(K\) is denoted by \(K(x;\sigma,\delta)\). The paper under review studies the structure of \(R\) from the perspective of order theory. Note that an order in a simple Artinian ring \(Q\) means a prime Goldie ring with a quotient ring \(Q\). Let \(S\) be an order in the ring \(Q\). A left \(S\)-submodule \(I\) of \(Q\) is said to be a left \(S\)-ideal, if \(I\) contains a regular element in \(S\) and \(Iu\subseteq S\), for some \(u\in U(Q)\), where \(U(Q)\) is the group of units in \(Q\). When this holds, \(I\) is called integral, if \(I\subseteq S\). Right \(S\)-ideals are defined similarly. A right and left \(S\)-ideal is said to be an \(S\)-ideal. Let \(I\) be a left \(S\)-ideal. Then \((S:I)_r=\{q\in Q:Iq\subseteq S\}\) is a right \(S\)-ideal and \(_vI=(S:(S:I)_r)_l\) is a left \(S\)-ideal including \(I\), where \((S:J)_l=\{q\in Q:qJ\subseteq S\}\) for any right \(S\)-ideal \(J\). An order \(S\) in \(Q\) is called a left \(v\)-Bezout order, if \(_vI\) is principal, for any finitely-generated left integral \(S\)-ideal \(I\) (a right \(v\)-Bezout order is defined similarly). We say that \(S\) is a \(v\)-Bezout order, if it is a left and right \(v\)-Bezout order; \(S\) is called a Bezout order, if every finitely-generated one-sided \(S\)-ideal is principal. Clearly, Bezou…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16H10
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 5818581
zbMATH Keywords: Ore extensions
zbMATH Keywords: total valuation rings
zbMATH Keywords: commutative GCD domains
zbMATH Keywords: fully left bounded rings of division rings
zbMATH Keywords: Bezout orders
MaRDI profile type: Publication
full work available at URL: s10468-009-9139-4
OpenAlex ID: W2044419484
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DOI: 10.1007/S10468-009-9139-4
description: scientific article; zbMATH DE number 5818581