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!



A nonreduced quotient ring
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.


