I have a question on the use of the chinese remainder theorem with regards to a ring isomorphism.
Suppose R is a ring and R = $\mathbb{R}[x]/(x^2-3x +2)$.
Then note that $(x^2 - 3x + 2) = (x - 1)(x-2)$. Set $I = (x-1)$ and $J = (x-2)$.
Now I + J = $\mathbb{R}[x]$, so $IJ$ = $I$ $\bigcap$ $J$.
Then we have the homomorphism $\phi$: $\mathbb{R}[x]$ -> ($\mathbb{R}[x]$ mod $I$, $\mathbb{R}[x]$ mod $J$) via the canonical projection.
Then the kernel of $\phi$ is I $\bigcap$ J, however I'm having trouble concluding that the kernel equals $(x-1)(x-2)$. I realize when $I$ and $J$ are comaximal that IJ = I $\bigcap$ J. But $IJ$ is not simply the product of (x-1) and (x-2). It is the set of all sums of products of elements in I and J.
Why is it that we can conclude $I$ $\bigcap$ $J$ = $(x-1)(x-2)$?
Thanks!