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Commutators and Anti-Commutators of Idempotents in Rings
- Source :
- AMS Contemporary Mathematics Volume 715, 2018, 205-224
- Publication Year :
- 2018
-
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 $\,e^2=e$.<br />Comment: 21 pages
- Subjects :
- Mathematics - Rings and Algebras
12E15, 15B33, 15B36, 16E50, 16N40, 16U60
Subjects
Details
- Database :
- arXiv
- Journal :
- AMS Contemporary Mathematics Volume 715, 2018, 205-224
- Publication Type :
- Report
- Accession number :
- edsarx.1808.02308
- Document Type :
- Working Paper