Let $R$ be a finitely generated commutative ring and $C$ an $R$-algebra ($C$ is not necessarily commutative). Assume that $C$ is a finitely generated $R$-module.
If $S$ is a simple $C$-module, then is the annihilator $I=Ann_{C}(S)$ of $S$ is of the form $I=\mathfrak{m}C$ for some maximal ideal $\mathfrak{m}$ of $R$?
Best Answer
Consider the case $R$ is a field $k$ (a finite field, if you really want it to be finitely generated as a commutative ring, although this plays no role in what follows, as far as I can see).
Then $\mathfrak m$ necessarily equals zero, and so the statement you ask about becomes: if $C$ is a finite dimensional $k$-algebra, and $S$ is a simple $C$-module, then $S$ is faithful (i.e. has trivial annihilator). This is false (you can easily construct counterexamples).