• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1179-1187
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• 2018

We study the class of rings R with the property that for $x{\in}R$ at least one of the elements x and 1 + x are tripotent. We prove that a commutative ring has this property if and only if it is a subring of a direct product $R_0{\times}R_1{\times}R_2$ such that $R_0/J(R_0){\cong}{\mathbb{z}}_2$, for every $x{\in}J(R_0)$ we have $x^2=2x$, $R_1$ is a Boolean ring, and $R_3$ is a subring of a direct product of copies of ${\mathbb{z}}_3$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.47-58
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• 2021

In this article, we completely determine the rings for which every element is a sum of a nilpotent and three tripotents that commute with one another. We discuss this property for some extensions of rings, including group rings.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.197-224
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• 2019

Let A be a $JB^*$-triple with Banach dual space $A^*$ and bi-dual the $JBW^*$-triple $A^{**}$. Elements x of $A^*$ of norm one may be regarded as normalised 'Q-measures' defined on the complete ortho-lattice ${\tilde{\mathcal{U}}}(A^{**})$ of tripotents in $A^{**}$. A Q-measure x possesses a support e(x) in ${\tilde{\mathcal{U}}}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice ${\tilde{\mathcal{U}}}_c(A)$ of elements of ${\tilde{\mathcal{U}}}(A^{**})$ compact relative to A. Necessary and sufficient conditions for an element v of ${\tilde{\mathcal{U}}}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of v by families of elements of ${\tilde{\mathcal{U}}}_c(A)$.