Open AccessON THE COMMUTATIVITY AND ANTICOMMUTATIVITY OF RINGS II
Author(s) -
Chen-Te Yen
Publication year - 1991
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.22.1991.4610
Subject(s) - mathematics , nilpotent , commutative property , commutative ring , ring (chemistry) , combinatorics , associative property , natural number , discrete mathematics , pure mathematics , chemistry , organic chemistry
It is shown that if $R$ is any associative ring such that for each $x,y\in R$, there exist an even natural number $m(x,y)$ and an odd natural number $n(x,y)$, depending on $x$ and $y$, with either $[x,y]^{m(x,y)} = [x,y]^{n(x,y)}$ or $(x\circ y)^{m(x, y)} = (x \circ y)^{n(x,y)}$, then either $[x,y]$ or $(x \circ y)$ is nilpotent for all $x, y$ in $R$. Moreover, $R$ is commutative if $R$ has no nonzero nil right ideals.