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Maximal commutative subrings and simplicity of Ore extensions

Öinert, Johan LU ; Richter, Johan LU and Silvestrov, Sergei LU (2013) In Journal of Algebra and Its Applications 12(4). p.16-1250192
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 \delta-simple. 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 non-zero ideal of R[x;id_R,\delta] non-trivially. Using this we show that if R is \delta-simple 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:
author
; and
organization
publishing date
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
0219-4988
DOI
10.1142/S0219498812501927
language
English
LU publication?
yes
id
a2a2cc23-d425-487b-9a7d-4fb1e79ca27f (old id 3409325)
alternative location
http://arxiv.org/abs/1111.1292
date added to LUP
2016-04-01 10:13:01
date last changed
2022-04-12 03:07:28
@article{a2a2cc23-d425-487b-9a7d-4fb1e79ca27f,
  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 \delta-simple. 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 non-zero ideal of R[x;id_R,\delta] non-trivially. Using this we show that if R is \delta-simple 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         = {{0219-4988}},
  keywords     = {{Ore extension rings; maximal commutativity; ideals; simplicity}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{16--1250192}},
  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}},
}