Rings in Which Every Element is a Sum of Two Tripotents

Canadian mathematical bulletin

Tamer Ko?an³  Zhiling Ying¹  Yiqiang Zhou² 

[1] College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P.R. China;Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, NL A1C 5S7, Canada;Department of Mathematics, Gebze Technical University, Gebze/Kocaeli, Turkey

关键词: idempotent;    tripotent;    Boolean ring;    polynomial identity $x^3=x$;    polynomial identity $x^6=x^4$;    polynomial identity $x^8=x^4$;   

DOI  :  10.4153/CMB-2016-009-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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 $Rcong R_1imes R_2$, 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 $Rcong R_1imes R_2$, 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 $Rcong R_1imes R_2imes R_3$, 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 subdirectproduct of $mathbb Z_5$'s.

【 授权许可】

Unknown