I am currently reading the book Heat Kernels and Dirac Operators and I am facing the following problem:
Let $S$ be a complex and finite-dimensional vector space and $E$ a finite-dimensional $\mathrm{End}(S)$-module. The proof of proposition $3.27$ says that this means that
$E$ is of the form $S\otimes W$.
It sounds like they are referring to some standard result from algebra, can someone please elaborate and/or name a reference?
Edit: I guess that what they really mean is that we can find a complex vector space $W$ together with an isomorphism $E\to S\otimes W$ of $\mathrm{End}(S)$-modules , where the action of $\mathrm{End}(S)$ on $S\otimes W$ is simply given by $$A\cdot(s\otimes w)=A(s)\otimes w.$$ What do you think?
A ring theorist would say that $R=End(S)$ is a semisimple ring, and as such, $S$ is the unique simple module of $R$.
Since every $R$ module is a direct sum simple modules (that's a characterizing property of semisimple rings) and the only simple module is $S$, $E$ has to be a sum of copies of $S$, and its dimensionality would wind up being some multiple of $\dim(S)$, or else some infinite cardinal.
If the dimension of $E$ is $n\dim(S)$, then you can pick $W$ to be any $n$ dimensional space so that $S\otimes_\mathbb C W$ has the right dimension to be isomorphic (as vector spaces) to $E$. If the dimension of $E$ is some infinite cardinal, then you can just pick $W$ to have the same dimension as $E$.
The $R$ module structure it gets from $\oplus_{i\in I} S$ can be given to $S\otimes_\mathbb C W$, but I'm not sure why it's any more useful than regarding it as $\oplus_{i\in I} S$. The action you described does make $S\otimes_\mathbb C W$ into a left $R$ module, and owing to the dimensionality it would be isomorphic to the direct sum.