Classification of rings satisfying some constraints on subsets

Khan, Moharram A.. "Classification of rings satisfying some constraints on subsets." Archivum Mathematicum 043.1 (2007): 19-29. <http://eudml.org/doc/250180>.

@article{Khan2007,
abstract = {Let $R$ be an associative ring with identity $1$ and $J(R)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\backslash J(R)$ and (ii) $[x,[x,y^m]]=0$, for all $x,y\in R\backslash J(R)$. This result is also valid if (ii) is replaced by (ii)’ $[(yx)^mx^m-x^m(xy)^m,x]=0$, for all $x,y\in R\backslash N(R)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).},
author = {Khan, Moharram A.},
journal = {Archivum Mathematicum},
keywords = {Jacobson radical; nil commutator; periodic ring; Jacobson radical; nil commutators; periodic rings; commutativity theorems; commutator constraints},
language = {eng},
number = {1},
pages = {19-29},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Classification of rings satisfying some constraints on subsets},
url = {http://eudml.org/doc/250180},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Khan, Moharram A.
TI - Classification of rings satisfying some constraints on subsets
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 1
SP - 19
EP - 29
AB - Let $R$ be an associative ring with identity $1$ and $J(R)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\backslash J(R)$ and (ii) $[x,[x,y^m]]=0$, for all $x,y\in R\backslash J(R)$. This result is also valid if (ii) is replaced by (ii)’ $[(yx)^mx^m-x^m(xy)^m,x]=0$, for all $x,y\in R\backslash N(R)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).
LA - eng
KW - Jacobson radical; nil commutator; periodic ring; Jacobson radical; nil commutators; periodic rings; commutativity theorems; commutator constraints
UR - http://eudml.org/doc/250180
ER -