Simple rings and degree maps
(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)
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
- Academic Press
- 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
- 2016-04-01 10:15:18
- date last changed
- 2025-10-14 12:25:54
@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 $d(a'b - ba') < d(a)$ 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}},
keywords = {{simplicity; degree map; ring extension; ideal associativity}},
language = {{eng}},
pages = {{201--219}},
publisher = {{Academic Press}},
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}},
}