Maximal commutative subrings and simplicity of Ore extensions
(2013) In Journal of Algebra and Its Applications 12(4). p.161250192 Abstract
 The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id_R,\delta], is simple if and only if its center is a field and R is \deltasimple. When R is commutative we note that the centralizer of R in R[x;\sigma,\delta] is a maximal commutative subring containing $R$ and, in the case when \sigma=id_R, we show that it intersects every nonzero ideal of R[x;id_R,\delta] nontrivially. Using this we show that if R is \deltasimple and maximal commutative in R[x;id_R,\delta], then R[x;id_R,\delta] is simple. We also show that under some conditions on R the converse holds.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3409325
 author
 Öinert, Johan ^{LU} ; Richter, Johan ^{LU} and Silvestrov, Sergei ^{LU}
 organization
 publishing date
 2013
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Ore extension rings, maximal commutativity, ideals, simplicity
 in
 Journal of Algebra and Its Applications
 volume
 12
 issue
 4
 pages
 16  1250192
 publisher
 World Scientific Publishing
 external identifiers

 wos:000316952300011
 scopus:84874390665
 ISSN
 02194988
 DOI
 10.1142/S0219498812501927
 language
 English
 LU publication?
 yes
 id
 a2a2cc23d425487b9a7d4fb1e79ca27f (old id 3409325)
 alternative location
 http://arxiv.org/abs/1111.1292
 date added to LUP
 20160401 10:13:01
 date last changed
 20220412 03:07:28
@article{a2a2cc23d425487b9a7d4fb1e79ca27f, abstract = {{The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id_R,\delta], is simple if and only if its center is a field and R is \deltasimple. When R is commutative we note that the centralizer of R in R[x;\sigma,\delta] is a maximal commutative subring containing $R$ and, in the case when \sigma=id_R, we show that it intersects every nonzero ideal of R[x;id_R,\delta] nontrivially. Using this we show that if R is \deltasimple and maximal commutative in R[x;id_R,\delta], then R[x;id_R,\delta] is simple. We also show that under some conditions on R the converse holds.}}, author = {{Öinert, Johan and Richter, Johan and Silvestrov, Sergei}}, issn = {{02194988}}, keywords = {{Ore extension rings; maximal commutativity; ideals; simplicity}}, language = {{eng}}, number = {{4}}, pages = {{161250192}}, publisher = {{World Scientific Publishing}}, series = {{Journal of Algebra and Its Applications}}, title = {{Maximal commutative subrings and simplicity of Ore extensions}}, url = {{http://dx.doi.org/10.1142/S0219498812501927}}, doi = {{10.1142/S0219498812501927}}, volume = {{12}}, year = {{2013}}, }