Locally nilpotent derivations on modules
type: Item
instance of: scholarly article
title: Locally nilpotent derivations on modules
zbMATH Open document ID: 1180.14057
author: Mikiya Tanaka
published in: Journal of Mathematics of Kyoto University
publication date: 2009-09-01T00:00:00Z
review text: Let \(B\) be a ring and \(\partial\) be a locally nilpotent derivation on \(B\), and let \(A:=\ker(\partial)\). Then \((M,\partial_M)\) is defined as a pre-\(\partial\)-module if \(M\) is a \(B\)-module and \(\partial_M\) satisfies \(\partial_M(bm)=\partial(b)m+b\partial_M(m)\) for each \(b\in B, m\in M\). If \(\delta_M\) is locally nilpotent, then the prefix ``pre'' is removed. The module \(M_0:=\ker(\partial_M)\) of a \(\delta\)-module \(M\) is studied, in particular its (non)finite generation as a (1) \(B\)- or (2) \(A\)-module. Amongst others, the author gives affirmative answers to (1) in case \(M\) is a free \(B\)-module of rank one or if \(\partial\) has a slice. A negative answer to (1) is given for the case that \(M\) is a projective \(B\)-module of rank one. The problem (2) is affirmatively answered if \(M\) is a free \(B\)-module of rank one, or if \(\partial\) has a slice.
reviewed by: Stefan Maubach
Mathematics Subject Classification ID: 14R20
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5601042
zbMATH Keywords: locally nilpotent derivation
zbMATH Keywords: additive group action
zbMATH Keywords: module
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1248983033
OpenAlex ID: W1531185037
DOI: 10.1215/KJM/1248983033
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description: scientific article; zbMATH DE number 5601042