  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 $\,e^2=e$.
