How can a commutative algebra have more than one simple submodule? A result from representation theory:
-Let A be a commutative algebra over $\mathbb{C}$. Any simple A-module is a one-dimensional module over $\mathbb{C}$.
Is the following true or false? "A commutative group algebra $\mathbb{C}G$ can only have one simple submodule"
I feel like it should be true, as if we have a non-zero generating element "a" for one such submodule A, then since A is one-dimensional, ba=$\lambda$a for all b in $\mathbb{C}G$. An element "a" with such a property should be unique.
If the above statement is true, then this is where my confusion lies:
Consider $\mathbb{C}G$, where $G=C_{\large 3}$. This is a commutative group algebra. As a module over itself, it is 3 dimensional, and by the above reasoning, every simple submodule should be one-dimensional. 
I can find one one-dimensional submodule, $S_1 = $ $ <e + g + g^2>$, and one two-dimensional submodule, $S_2 = $ $<e - g, g - g^2>$, but I can't find a submodule of $S_2$, so it appears as if $S_2$ is a simple, two dimensional submodule of a commutative algebra. 
If anyone can help me at all it will be much appreciated!

In your example, you could have $S'=\left<e+\omega g+\omega^2 g^2\right>$
where $\omega=\exp(2\pi i/3)$. This is a one-dimensional module but not
isomorphic to $S_1$.
