Is a ring with the following properties semiprime? Let $R$ be a ring with $1 \neq 0$ that contains a noncentral idempotent $e$, and let $f = 1-e$.  If the corner rings $eRe$ and $fRf$ are both division rings and $eRf$ and $fRe$ are both nonzero, is the ring $R$ semiprime?

My example to your other question is a ring satisfying your conditions with non-zero Jacobson radical, which is a nilpotent right ideal.
