Homomorphism from $\mathbb R^2\to \mathbb C$

Question

Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows:

$(a,b)(c,d)=(ac,bd)$

Answer 1

Hint: If $A$ and $B$ are rings (with $1$), then the ideals of $A\times B$ are exactly the subsets in the form $I\times J$ for some pair of ideals $I\subseteq A$ and $J\subseteq B$. Thus...