Showing that $(\mathbb{R},+,\cdot)$ and $(\mathcal{M}_2(\mathbb{R}),+,\cdot)$ are not isomorphic rings Prove that $(\mathbb{R},+,\cdot)$ and $(\mathcal{M}_2(\mathbb{R}),+,\cdot)$ are not isomorphic rings.
I came up with the following argument, but I am not sure it works : the equation $x^2=1$ has only two solutions in $\mathbb{R}$, whereas in $\mathcal{M}_2(\mathbb{R})$ it has infinitely many (for instance, any matrix $A=\left(\begin{matrix} 1 & 0 \\ a & -1 \\ \end{matrix} \right)$ where $a$ is an arbitrary real number works). Is this enough to conclude that the two rings are not isomorphic?
EDIT: Thanks everyone and sorry for forgetting to type that minus sign !

One is commutative;  the other isn't. 
