Skew power series rings and derivations
type: Item
instance of: scholarly article
title: Skew power series rings and derivations
zbMATH Open document ID: 0542.16035
DOI: 10.1016/0021-8693(84)90143-1
author: H. H. Brungs
author: Günter Törner
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: The authors study skew power series rings in z over a skew field F with the commutation rule \(az=za^{\delta_ 0}+z^ 2a^{\delta_ 1}+...,\) where multiplication preserves the order, thus \(a^{\delta_ 0}=0\) implies \(a=0\). The appropriate sequence \((\delta_ 0,\delta_ 1,...)\) (here called ''admissible'', but also ''higher \(\delta_ 0\)-derivations'') consists of additive mappings of F satisfying an analogue of Leibniz's formula. For a commutative field F of characteristic 0 it is shown that in any admissible sequence \((1,\delta_ 1,\delta_ 2,...)\) the terms \(\delta_ 1,...,\delta_{n-1}\) determine \(\delta_ n\) up to a derivation of F. A particular case is that where the extra term is \(a_ n\delta\) for \(a_ n\in F\) and this leads to polynomials \(\delta_ n=g_ n(\delta)\) whose coefficients depend on \(a_ 1,...,a_ n\). For any a's in the kernel of \(\delta\) this leads to an admissible sequence \((1,g_ 1(\delta),g_ 2(\delta),...),\) but it is not known whether this is admissible for any choice of a's in F. It is shown that for any derivation \(\delta\) and any integer \(k\geq 1\), the definition \(\delta_ i=0\) for \(k\nmid j\), \(\delta_{nk}=(1/n!)(1+k)(1+2k)...(1+(n- 1)k)\delta^ k\) yields an admissible sequence.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 3861383
zbMATH Keywords: skew power series rings
zbMATH Keywords: commutation rule
zbMATH Keywords: admissible sequence
zbMATH Keywords: derivation
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90143-1
OpenAlex ID: W2050302061
cites work: On the Structure and Ideal Theory of Complete Local Rings
cites work: Q5639812
cites work: Formal Power Series
cites work: Q5536687
cites work: Q4193608
description: scientific article; zbMATH DE number 3861383