Tensor product of quotient rings $A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?

Define 
$$\phi : A/a \times A/b \rightarrow A/(a+b),\; \phi(x+a,y+b) = xy + (a+b).$$ 
Can you show $\phi$ is a well-defined bilinear map? 
Now if you use the universal property of tensor products to get a map 
$$\psi : A/a \otimes_A A/b \rightarrow A/(a+b),$$ can you show $\psi$ is injective? To show surjectivity, it will probably help you to show that an arbitrary simple tensor in $A/a \otimes_A A/b$ can be written in the form $(1+a)\otimes(x+b)$...
