Commutators and Anti-Commutators of Idempotents in Rings

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 $\,e^2=e$.Comment: 21 page

HOMFLY-PT skein module of singular links in the three-sphere

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links