How can one prove that any finite dimensional commutative $\mathbb{R}$-algebra $A$ has simple modules of dimension $\le$ 2

Question

How can one prove that any finite dimensional commutative $\mathbb{R}$-algebra $A$ has simple modules of dimension $\le 2$. I understand why the analagous result holds for $\mathbb{C}$ and dimension $1$, however I am struggling to relate this to the $\mathbb{R}$ case.

Answer 1

The simple modules of $A$ coincide with those of $A/J(A)$ where $J()$ means the Jacobson radical. Without loss of generality then, $A$ is semisimple Artinian.

The Artin-Wedderburn theorem says that $A$ has to be a finite product of matrix rings over finite dimensional division ring extensions of $\mathbb R$, but since $A$ was assumed commutative, the matrix rings all have to be $1\times 1$ and the division rings are in fact finite field extensions of $\mathbb R$.

There are only two of these, $\mathbb R$ and $\mathbb C$.

So the ring is actually a direct sum of copies of $\mathbb R$ and $\mathbb C$. A quotient by a maximal ideal therefore (which is how all simple modules arise) makes something of $\mathbb R$ dimension $1$ or $2$.