Simple modules of a finite dimensional commutative $\mathbb{R}$-algebra

Question

From Schur's Lemma, I know that simple modules of a finite dimensional commutative $\mathbb{C}$-algebra are one-dimensional over $\mathbb{C}$. But what can we say about the dimension of the simple modules when they are over $\mathbb{R}$? I'd have to assume their dimensions would be $\leq 2$, but how would I show this?

Answer 1

Simple modules over a ring $A$ are just quotients of $A$ by maximal left ideals. When $A$ is commutative, this means simple modules are just quotient rings of $A$ which are fields.

In particular, if $A$ is a finite-dimensional commutative algebra over a field $k$, every simple module is a field which is a finite extension of $k$. For $k=\mathbb{R}$, the only such fields are $\mathbb{R}$ and $\mathbb{C}$. So any simple module over a finite-dimensional commutative $\mathbb{R}$-algebra has dimension $1$ or $2$ over $\mathbb{R}$.