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# Rings in Which Every Element is a Sum of Two Tripotents

Published:2016-03-14
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
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## 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 $x^8=x^4$
 MSC Classifications: 16S50 - Endomorphism rings; matrix rings [See also 15-XX] 16U60 - Units, groups of units 16U90 - unknown classification 16U90

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