TITLE: Tensor product of $\mathbb R$ and $\mathbb C$ over $\mathbb R$.
QUESTION [7 upvotes]: $$ \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} = \;? $$
I guess this guy is just $\mathbb{C}$, is this correct?

REPLY [2 votes]: You could use that you have a (natural) isomorphism $M \cong M \otimes_R R$, see here for a proof.
Then $M = \mathbb C$ is an $R$-module for $R = \mathbb R$ and you get (as you suspected) that $\mathbb C \cong \mathbb R \otimes_{\mathbb R} \mathbb R$.