Simple rings and degree maps
(2014) In Journal of Algebra 401. p.201219 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 nontrivial 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 nonnegative 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 nontrivial 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 nonnegative 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 nonzero ideal $I$ of $A$ and each nonzero $a \in I$ there is a nonzero $a' \in I$ 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 CayleyDickson doublings. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4194508
 author
 Nystedt, Patrik and Öinert, Johan ^{LU}
 organization
 publishing date
 2014
 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
 00218693
 DOI
 10.1016/j.jalgebra.2013.11.023
 language
 English
 LU publication?
 yes
 id
 ea7da931082c4756993020930d9193dc (old id 4194508)
 date added to LUP
 20160401 10:15:18
 date last changed
 20210630 04:30:23
@article{ea7da931082c4756993020930d9193dc, 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 nontrivial 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 nonnegative 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 nonzero ideal $I$ of $A$ and each nonzero $a \in I$ there is a nonzero $a' \in I$ 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 CayleyDickson doublings.}, author = {Nystedt, Patrik and Öinert, Johan}, issn = {00218693}, language = {eng}, pages = {201219}, 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}, }