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Simple rings and degree maps

Nystedt, Patrik and Öinert, Johan LU (2014) In Journal of Algebra 401. p.201-219
Abstract
For an extension $A/B$ of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of $A$ with a property that we call $A$-simplicity of $B$. By this we mean that there is no non-trivial ideal $I$ of $B$ being $A$-invariant, that is satisfying $AI \subseteq IA$. We show that $A$-simplicity of $B$ is a necessary condition for simplicity of $A$ for a large class of ring extensions when $B$ is a direct summand of $A$. To obtain sufficient conditions for simplicity of $A$, we introduce the concept of a degree map for $A/B$. By this we mean a map $d$ from $A$ to the set of non-negative integers satisfying the following two conditions: (d1) if $a \in A$, then $d(a) = 0$ if and only if $a=0$;... (More)
For an extension $A/B$ of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of $A$ with a property that we call $A$-simplicity of $B$. By this we mean that there is no non-trivial ideal $I$ of $B$ being $A$-invariant, that is satisfying $AI \subseteq IA$. We show that $A$-simplicity of $B$ is a necessary condition for simplicity of $A$ for a large class of ring extensions when $B$ is a direct summand of $A$. To obtain sufficient conditions for simplicity of $A$, we introduce the concept of a degree map for $A/B$. By this we mean a map $d$ from $A$ to the set of non-negative integers satisfying the following two conditions: (d1) if $a \in A$, then $d(a) = 0$ if and only if $a=0$; (d2) there is a subset $X$ of $B$ generating $B$ as a ring such that for each non-zero ideal $I$ of $A$ and each non-zero $a \in I$ there is a non-zero $a' \in I$ with $d(a') \leq d(a)$ and $d(a'b - ba') < d(a)$ for all $b \in X$. We show that if the centralizer $C$ of $B$ in $A$ is an $A$-simple ring, every intersection of $C$ with an ideal of $A$ is $A$-invariant, $A C A = A$ and there is a degree map for $A/B$, then $A$ is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
simplicity, degree map, ring extension, ideal associativity
in
Journal of Algebra
volume
401
pages
201 - 219
publisher
Elsevier
external identifiers
  • wos:000330599500011
  • scopus:84891812645
ISSN
0021-8693
DOI
10.1016/j.jalgebra.2013.11.023
language
English
LU publication?
yes
id
ea7da931-082c-4756-9930-20930d9193dc (old id 4194508)
date added to LUP
2014-01-31 19:50:34
date last changed
2017-07-23 03:11:21
@article{ea7da931-082c-4756-9930-20930d9193dc,
  abstract     = {For an extension $A/B$ of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of $A$ with a property that we call $A$-simplicity of $B$. By this we mean that there is no non-trivial ideal $I$ of $B$ being $A$-invariant, that is satisfying $AI \subseteq IA$. We show that $A$-simplicity of $B$ is a necessary condition for simplicity of $A$ for a large class of ring extensions when $B$ is a direct summand of $A$. To obtain sufficient conditions for simplicity of $A$, we introduce the concept of a degree map for $A/B$. By this we mean a map $d$ from $A$ to the set of non-negative integers satisfying the following two conditions: (d1) if $a \in A$, then $d(a) = 0$ if and only if $a=0$; (d2) there is a subset $X$ of $B$ generating $B$ as a ring such that for each non-zero ideal $I$ of $A$ and each non-zero $a \in I$ there is a non-zero $a' \in I$ with $d(a') \leq d(a)$ and $d(a'b - ba') &lt; d(a)$ for all $b \in X$. We show that if the centralizer $C$ of $B$ in $A$ is an $A$-simple ring, every intersection of $C$ with an ideal of $A$ is $A$-invariant, $A C A = A$ and there is a degree map for $A/B$, then $A$ is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings.},
  author       = {Nystedt, Patrik and Öinert, Johan},
  issn         = {0021-8693},
  keyword      = {simplicity,degree map,ring extension,ideal associativity},
  language     = {eng},
  pages        = {201--219},
  publisher    = {Elsevier},
  series       = {Journal of Algebra},
  title        = {Simple rings and degree maps},
  url          = {http://dx.doi.org/10.1016/j.jalgebra.2013.11.023},
  volume       = {401},
  year         = {2014},
}