Let $I$ be left ideal and $J$ be two-sided ideal in $R$. Prove if $I, J$ are nil then $I + J$ is nil left ideal. I even got a hint: consider $(I + J)/J$ in $R/J$, but I still don't know how to prove it.
edit: well, I can at least prove that $I + J$ is left ideal.

Hint for your hint: 
The idea is to show that $(I+J)/J$ is a nil left ideal of $R/J$.  This means that for any $i\in I$, $j\in J$, there is a power of $i+j$ in $J$. 
You can see the last step after that, right?
