Rings with a setwise polynomial-like condition

Document Type: Research Paper

Authors

1 Islamic Azad University - Majlesi Branch

2 University of Isfahan

3 Brock University

Abstract

Let $R$ be an infinite ring. Here we
prove that if $0_R$ belongs to ${x_1x_2cdots x_n ;|;
x_1,x_2,dots,x_nin X}$ for every infinite subset $X$ of $R$,
then $R$ satisfies the polynomial identity $x^n=0$. Also we prove
that if $0_R$ belongs to ${x_1x_2cdots
x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in X}$ for every
infinite subset $X$ of $R$, then $x^n=x$ for all $xin R$.

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Main Subjects