Problem: Let $R$ be a commutative ring with unity. Let $I$ and $J$ be coprime ideals of $R$. Prove $I \oplus J \cong R \oplus IJ$ as $R$-modules.
Attempt: I know that $I + J = (1) = R$. I wanted to construct the map $$ \phi: I \oplus J \rightarrow R \oplus IJ : (x,y) \mapsto (x+y, xy). $$
I think it is easy to see it is injective. I wanted to show it is surjective. A general element in the codomain looks like $(r, \sum_{i=1}^n x_i y_i)$ where $r \in R$, $x_i \in I$ and $y_i \in J$. Then I don't know how to prove that this element is actually in the image of $\phi$.
Help/advice is appreciated.
Consider for any coprime left ideals $I, J ⊆ R$ the exact sequence of $R$-modules $$0 → I∩J → I \oplus J → R → 0,$$ given by $$I∩J → I \oplus J,~x ↦ (x,-x) \quad\text{and}\quad I \oplus J → R,~(x,y) ↦ x + y.$$ Note that $IJ ⊇ (I+J)(I∩J)$ in general (so $IJ = I∩J$ in this case) and that $R$ is a free $R$-module, making this sequence split.