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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 $C=R$?2012-08-15
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    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.2012-08-15

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