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arXiv · 1808.02308

Commutators and Anti-Commutators of Idempotents in Rings

Dinesh Khurana·T. Y. Lam
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Abstract

We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the "anti-commutator" $\,ee'+e'e\,$ is automatically invertible, so we study also the broader class of rings having such an invertible anti-commutator. Simple artinian rings $\,R\,$ (along with other related classes of matrix rings) with one of the above properties are completely determined. In this study, we also arrive at various new criteria for {\it general\} $\,2\times 2\,$ matrix rings. For instance, $R\,$ is such a matrix ring if and only if it has an invertible commutator $\,er-re\,$ where $\,e^2=e$.

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Dinesh Khurana, T. Y. Lam. 2018-08-07. Commutators and Anti-Commutators of Idempotents in Rings. https://arxiv.org/abs/1808.02308

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