I apologize if this is a duplicate, but I haven't found proper answers in the forums.
It is easy to show that the quaternions and $M_2(\mathbb R)$ are isomorphic as vector spaces over the reals, but I need to show that they are not ring-isomorphic.
I thought maybe to assume some $\varphi$ is a ring-homomorphism and get a contradiction, similar to a typical proof showing $2\mathbb Z$ is not isomorphic to $3\mathbb Z$ for example. But here I find it trickier to get a contradicting example and perhaps this isn't the right approach.
Could the isomorphism theorems be useful here? Or how can it be shown otherwise? Thanks.