Matrix Algebras

Question

I'm reading the papaer "On the Reduction of a Matrix to Diagonal Form" of Epstein and Flanders (Amer. Math. Monthly 62, (1955). 168–171.

Let $S$ denote the trace function.

The authors stated that a well-known result in the theory of algebras of matrices is:

A matrix algebra $\mathbb U$ over a field $\mathbb F$ of characteristice zero is semisimples if and only if $S(XY)=0$, for fixed $X\in\mathbb U$ and all $Y\in\mathbb U$,implies $X=0$.

The authors do not give references. Does anyone know where I can find a proof of this result?

Answer 1

Any book on Ring Theory covering Artin-Wedderburn's Theorem :-))

To be fair, you need some dexterity in using it:

If $U$ is not semisimple, then it has nonzero nilpotent ideal $I$. All elements $x\in I$ will have zero trace since $x^n=0$. Hence $S(Iy)=0$ for any $y$.

In the opposite direction, let $I$ be the kernel of the form $<x,y>=S(xy)$. Clearly, $I$ is an ideal and $S(x^n)=0$ for all $x\in I$ and natural $n$. In characteristic zero, this means that such $x$ is nilpotent, i.e. $I$ is a nil ideal. It must be zero, since $U$ is semisimple.