TITLE: Show that $\mathbb{R}\oplus\mathbb{R}$ is not ring isomorphic to $\mathbb{C}$.
QUESTION [5 upvotes]: Show that $\mathbb{R}\oplus\mathbb{R}$ is not ring isomorphic to $\mathbb{C}$.  
This is my first abstract algebra class and therefore if the explanation could be kept as simple as possible that would be very much appreciated. I dont even know how to approach this problem.

REPLY [6 votes]: Nontrivial direct sums always have zero divisors, and $\mathbb C$ doesn't.
$\mathbb C$ has other algebraic properties that $\mathbb R\oplus \mathbb R$ doesn't have, like existence of solutions to polynomial equations, which would be preserved under ring isomorphism.  For example, $\mathbb R\oplus \mathbb R$ has no element whose square is $(-1,-1)$, whereas $z^2=-1$ does have solutions in $\mathbb C$.