Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed).
I would like to know what is the minimal possible $m$ such that it has an $m$-dimensional faithful representation.
Of course, $m$ is at most $n$, since any such algebra $A$ has a faithful representation $A\to End(A)$ given by left multiplication. And also it is clear that $m\geq \sqrt{n}$ by dimensional reasons.
If I remember correctly, for any finite dimensional algebra $A$ if $rad(A)$ denotes the maximal nilpotent $2$-sided ideal, then $A/rad(A)$ is a product of matrix algebras. For any matrix algebra it is clear what is such $m$. So I guess it reduces the problem to the case when for $A$ the ideal $rad(A)$ is maximal. So can we tell anything then?
I hope this is not a stupid question... I might be overlooking something very simple here.