Geodesic
Rings in which Every Element is a Sum of Two Tripotents
Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 661-672

Voir la notice de l'article provenant de la source Cambridge University Press

Let $R$ be a ring. The following results are proved. $\left( 1 \right)$ Every element of $R$ is a sum of an idempotent and a tripotent that commute if and only if $R$ has the identity ${{x}^{6}}\,=\,{{x}^{4}}$ if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$ . $\left( 2 \right)$ Every element of $R$ is either a sum or a difference of two commuting idempotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $J\left( R \right)\,=\,0$ or $J\left( R \right)\,=\,\left\{ 0,\,2 \right\}$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$ . $\left( 3 \right)$ Every element of $R$ is a sum of two commuting tripotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}\,\times \,{{R}_{3}}$ , where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ 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 ${{\mathbb{Z}}_{5}}^{,}s$ .
DOI : 10.4153/CMB-2016-009-0
Mots-clés : 16S50, 16U60, 16U90, idempotent, tripotent, Boolean ring, polynomial identity x 3 = x, polynomial identity x 6 = x 4, polynomial identity x 8 = x 4 
Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang. Rings in which Every Element is a Sum of Two Tripotents. Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 661-672. doi: 10.4153/CMB-2016-009-0
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