Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$
This is on a problem sheet for my class. I haven't a clue where to begin, this is the first isomorphism with a product that I've seen. All I've found so far is that there is some $a$ from $I$ and $b$ from $J$ such that $a + b = 1$ since $1$ is an element of $R = I + J$, and $I$ don't see how that helps.
For any $r,s\in R$, we can write $r=r_i+r_j$ and $s=s_i+s_j$, where $r_i,s_i\in I$, etc. Let $a=r_i+s_j$, so $a-r=s_j-r_j\in J$, and $a-s=r_i-s_i\in I$. In particular, $a+I=s+I$, and $a+J=r+J$. This shows that the map $$ \varphi\colon R\to R/I\times R/J:a\mapsto (a+I,a+J) $$ is surjective. Verify it is a homomorphism, find its kernel, and apply the isomorphism theorems to get the result.