TITLE: Error in proof: $\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$??
QUESTION [14 upvotes]: I've unintentionally "proved" the following:
$$\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$$
Can you help me tracing the error I made resulting to this non-proof? Here it is.
First of all, I recall an algebraic theorem about the complex plane: $\mathbb{C} \cong  \mathbb{R}[X]/(X^2+1)$. The rest of the non-proof is about the apparent isomorphism $\mathbb{C} \times \mathbb{C} \cong  \mathbb{R}[X]/(X^2+1)$.
The ring $R[X]$ is a commutative ring containing a unit, so we can write the ideal $(X^2+1)$ as a product of ideals $(X-i)(X+i)$. These ideals are indivisable: 
$(X+i)(X-i) \ni \frac{1}{2}(X+i)-\frac{1}{2}(X-i)=i$, so every polynomial 
$q = i^3q \cdot i$ is contained in the sum of ideals. Now we can use the generalisation of the Chinese remainder theorem: 
For a commutative ring R with unity, and indivisible ideals $I,J$ applies 
$R/IJ \cong R/I \times R/J$.
Now we have $\mathbb{R}[X]/(X^2+1) \cong \mathbb{R}[X]/(X+i) \times \mathbb{R}[X]/(X-i)$. I recall that for every homomorphism of rings $f: R \rightarrow S$ the isomorphism $R/\ker(f) \cong f(R)$ holds. Consider the mapping 
$s_r \ : \ R[X] \rightarrow S \ : \ \sum a_i X^i \mapsto \sum a_i r^i$. This substitution map is an homomorphism. A special case is the homomorphism $s_i \ : \ \mathbb{R}[X] \rightarrow \mathbb{C} \ : \ \sum a_i X^i \mapsto \sum a_i r^i$ that substitutes $i$ in $X$. Its kernel is exactly the ideal $(X-i)$. The image clairy contains $\{ a + bi: a,b \in \mathbb{R} \} = \mathbb{C}$, and the image is contained in $\mathbb{C}$ as well. So we obtain $\mathbb{C} \cong \mathbb{R}[X]/(X+i)$. The same trick with the substitution $s_{-i}$ shows that $\mathbb{C} \cong \mathbb{R}[X]/(X-i)$.
This results into $\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$. Here ends the "proof."
I feel a little bad about using $s_i$, because $i \notin \mathbb{R}$, but I remember that a similar mapping had to be used to proof that  $\mathbb{C} \cong  \mathbb{R}[X]/(X^2+1)$. 
I'd be thankful if you help me to sift this through and find the error(s).

REPLY [6 votes]: $ \Bbb R[x]/(x+i)$ is not defined as $x+i$ is not even in $ \Bbb R[x]$.