Ring homomorphism proof

Question

Let $I$ and $J$ be ideals of a ring $R$ and $I + J = R$. Show that $R/(I\cap J) \cong R/I \oplus R/J$.

So far my thought process has been to define a homomorphism $f:R \to R/I \oplus R/J$, and since the kernel of $f$ is $I\cap J$ we can use the fundamental theorem of ring homomorphisms. However, I still need to show that $f$ is surjective, which is where I'm stuck.

Answer 1

Hint:

Since $I+J=R$, we have $i+j=1$. Let $(a,b)\in R/I\times R/J$, then consider the element

$aj+bi$