# Annihilator of a simple module

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 $R$?

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What have you tried? Do you know anything about Jacobson radicals, or more specifically about versions of this problem with less hypotheses? In particular, the case $C=R$? –  Kevin Carlson Aug 15 '12 at 9:43
In case $R=C$ this is true; we have $S\cong R/\mathfrak{m}$ for some maximal ideal $\mathfrak{m}$ of $R$ as $S$ is simple. –  M. K. Aug 15 '12 at 10:08
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## 1 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).

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@M.K. If you need some inspiration for your counterexample, maybe you should check out primitive ring at Wikipedia. –  rschwieb Aug 18 '12 at 12:06
Sorry for my late reply. Now that I have better understanding on this problem. I appreciate your assistance. –  M. K. Aug 23 '12 at 5:56
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