Prove that $\mathbb{C}[x]/(x^2+2x)$ is isomorphic to $\mathbb{C} \oplus \mathbb{C} $

Question

Prove that $\mathbb{C}[x]/(x^2+2x)$ is isomorphic to $\mathbb{C} \oplus \mathbb{C} $, where $\mathbb{C} \oplus \mathbb{C} = \{(z_1, \space z_2) | z_1, z_2 \in \mathbb{C} \}$ is the ring with componentwise operations of addition and multiplication.

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Firstly, I would like to clarify what is $\mathbb{C} \oplus \mathbb{C} $ ? I don't quite understand that and therefore can't continue solving the problem.

But in general, I need to use the Isomorphism theorem for rings and thus find homomorphism $\phi$, such that $\operatorname{Ker} \phi = (x^2+2x)$ and $\operatorname{Im} \phi = \mathbb{C} \oplus \mathbb{C}$.

If $\phi : \mathbb{C}[x] \to \mathbb{C}$ and $f \mapsto f(0)$, then $\operatorname{Ker} \phi = (x)$. If it is $f \mapsto f(2)$, then $\operatorname{Ker} \phi = (x-2)$ and $\operatorname{Im} \phi = \mathbb{C}$. But how homomorphism $\phi$ should look like for Ker and Im to be as mentioned before?

Answer 1

The ring $\mathbb{C}\oplus\mathbb{C}$ (also written $\mathbb{C}\times\mathbb{C}$) has addition and multiplication defined by $$ (z_1,z_2)+(z_3,z_4)=(z_1+z_3,z_2+z_4), \qquad (z_1,z_2)(z_3,z_4)=(z_1z_3,z_2z_4) $$ Now consider your maps “tied together” (but with $-2$ rather than $2$): $$ \varphi\colon\mathbb{C}[x]\to\mathbb{C}\oplus\mathbb{C} \qquad \varphi(f)=(f(0),f(-2)) $$ and prove it's a ring homomorphism. What's its kernel?

Answer 2

Here is the ‘abstract’ version of the Chinese remainder theorem:

Le $R$ be a (commutative, with unit) ring, $I$ and $J$ two ideals of $R$ such that $I+J=R$. Then

1. $I\cap J=IJ$,
2. The canonical homomorphism $\begin{aligned}[t]R&\longrightarrow R/I\times R/J,\\ x&\longmapsto (x\bmod I,x\bmod J),\end{aligned}$

induces a ring isomorphism $\;R/I\cap J=R/IJ\longrightarrow R/I\times R/J$.