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Rings in Which Every Element is a Sum of Two Tripotents


Published:20160314
Printed: Sep 2016
Zhiling Ying,
College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P.R. China
Tamer Koşan,
Department of Mathematics, Gebze Technical University, Gebze/Kocaeli, Turkey
Yiqiang Zhou,
Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, NL A1C 5S7, Canada
Abstract
Let $R$ be a ring. The following results are proved: $(1)$ every
element of $R$ is a sum of an idempotent and a tripotent that
commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times
R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of
exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb
Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference
of two commuting idempotents iff $R\cong R_1\times R_2$, where
$R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$,
and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s;
$(3)$ every element of $R$ is a sum of two commuting tripotents
iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean
with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect
product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect
product of $\mathbb Z_5$'s.
Keywords: 
idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$
idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$
