Sum of nil left ideals

Question

Let $I,J$ be left ideals in a ring $A$ such that $\forall x\in I \ \ \forall y \in J$ we have that $x,y$ are nilpotent. Is then true that every element of $I+J$ is nilpotent? Now, if the ideals are bilateral, expanding $(x+y)^{n_0}$ and "playing" a little bit with the exponent $n_0$ we've proved that the answer is: yes. But I think a different approach is needed with the hypothesis that $I,J$ are left ideals. Any suggestion?