= Simple module of a finite-dimensional commutative complex algebra
{title2=$\dim_{\mathbb C}S=1$}
For a finite-dimensional commutative complex <algebra>, a <simple module> is a cyclic quotient of the algebra and hence finite-dimensional. Every action map is a module endomorphism, since the algebra is commutative. The complex <Schur's lemma> makes each action scalar, so every vector subspace is a submodule; simplicity forces dimension one. The finite-dimensional hypothesis cannot simply be deleted: $\mathbb C(t)$ acting on itself is a simple module of infinite complex dimension.
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