Let $R$ be a ring with $1 \neq 0$, let $e$ be any nontrivial idempotent in $R$, and let $f = 1-e$. Then we can write $$R \cong \begin{pmatrix} eRe & eRf \\ fRe & fRf \end{pmatrix}.$$ If the rings $eRe$ and $fRf$ are both division rings and $eRf$ and $fRe$ are both nonzero, is the ring $R$ simple?
Is a ring with the following properties simple?
1
$\begingroup$
ring-theory