# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Simple rings and degree maps

and (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)
author
and
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)
2016-04-01 10:15:18
date last changed
2021-06-30 04:30:23
@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},
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},
doi          = {10.1016/j.jalgebra.2013.11.023},
volume       = {401},
year         = {2014},
}