I would like to prove the following: let $A$ be a ring and $I,J\subset A$ two ideals. Then: $$A/I\otimes_AA/J\cong A/(I+J)$$ I have seen a proof using the Yoneda lemma (even though I haven't understood it fully yet), but I've been told there's also a proof using the short exact sequence: $$0\to I\to A\to A/I\to0$$ The only obvious thing to do here, is to apply the functor $-\otimes_AA/J$ to the sequence. This functor is right exact, and thus we get the exact sequence: $$I\otimes_AA/J\to A\otimes_AA/J\cong A/J\to A/I\otimes_AA/J\to0$$ Now my idea would be to try to show that the kernel of the homomorphism on the left is (isomorphic to) $(I+J)/J$, and then I would be done. However, I don't know how to show it.
Any advice? Alternatively, is there any other simple way to work this proof?