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I am searching for a commutative ring $R$ and a semisimple $R$-module $M$ such that the quotient ring $\frac {R}{soc(R)\cap ann_R(M)}$ has a nonzero nilpotent element. Here, $soc(R)$ means the socle of $R$ and $ann_R(M)$ stands for the annihilator of $M$ in $R$.

Thanks for any cooperation!

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Take $R=K[x]/(x^3)$ and $M=S$ the unique simple $R$-module. We have $soc(R)=S$ and $ann(M)=J$ the jacobson radical of $R$. Then $soc(R) \cap ann(M)=soc(R)$ and $R/soc(R) \cap ann(M)=R/soc(R)=K[x]/(x^2)$ has nonzero nilpotent element x.

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  • $\begingroup$ Thanks for the answer! Could one choose such $R$ and $M$ so that the intersection $soc(R)\cap ann_R(M)$ contains all the nilpotent elements of $R$? In fact, these are what I am searching for. Thanks again! $\endgroup$
    – karparvar
    Commented Mar 5, 2018 at 14:07

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