Let $F$ be a field and $A$ a $4$-dimensional $F$-algebra. Can I classify all such algebras as being either $\operatorname{Mat}_2(F)$ or a division algebra depending on whether they have zero divisors?
I'm not sure if the following is a proof. I think it depends on $A$ being a simple algebra, which I'm not sure if it comes for free from $\dim_F A =4$. Could someone verify or generalize?
If $A$ has a zero divisor, it can't be a division algebra. By Wedderburn's theorem, it is a matrix algebra over some division algebra $D$. But because the $\dim_F A = 2^2$, we necessarily have $\dim_F D =1$. So $A$ is a matrix algebra.
Conversely, Wedderburn's theorem still implies that $A=\operatorname{Mat}_n(D)$ even if $A$ doesn't have zero divisors. But then $n$ must be $1$, so $A$ is a division algebra.