There exists an exact sequence:
$\begin{aligned}
0 \rightarrow I \cap J \xrightarrow{f} I \oplus J \xrightarrow{g} I + J \rightarrow 0 \\
&f(r) = (r, r) \\
&g((i, j)) = i + j
\end{aligned}$

We are forced into this formula by considerations of dimension. We know:
$\begin{aligned}
&dim(I \oplus J) = dim(I) + dim(J) \\
&dim(I + J) = dim(I) + dim(J) - dim(I \cap J) \text{[inclusion-exclusion]} \\
&dim(I + J) = dim(I \oplus J) - dim(I \cap J) \\
&dim(I + J) - dim(I \oplus J) + dim(I \cap J) = 0\\
&V - E + F = 2
\end{aligned}$

By analogy to euler characteristic which arises from homology, we need to have
$I \oplus J$ in the middle of our exact sequence. So we must have:
$0 \rightarrow ? \rightarrow I \oplus J \rightarrow ?\rightarrow 0$

Now we need to decide on the relative ordering between $I \cap J$ and $I + J$.
- There is no
*universal* way to send $I oplus J \rightarrow I \cap J$. It'san unnatural operation to restrict the direct sum into the intersection. - There is a
*universal* way to send $I \oplus J \rightarrow I + J$: sumthe two components. This can be seen as currying the addition operation.

Thus, the exact sequence *must* have $I + J$ in the image of $I \oplus J$. This
forces us to arrive at:
$0 \rightarrow I \cap J \rightarrow I \oplus J \rightarrow I + J \rightarrow 0$

The product ideal $IJ$ plays no role, since it's not possible to define a
product of modules *in general* (just as it is not possible to define
a product of vector spaces). Thus, the exact sequence better involve
module related operations. We can now recover CRT:
$\begin{aligned}
0 \rightarrow I \cap J \xrightarrow{f} I \oplus J \xrightarrow{g} I + J \rightarrow 0 \\
0 \rightarrow R \xrightarrow{f} R \oplus R \xrightarrow{g} R \rightarrow 0 \\
0 \rightarrow R / (I \cap J) \rightarrow R/I \oplus R /J \rightarrow R/(I + J) \rightarrow 0
\end{aligned}$

#### § References