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I am searching for a ring $R$ and a semisimple $R$-module $M$ such that for any $r\in R$, either $r-r^2\in Soc(R_R)-ann_R^l(M)$ or $r+r^2\in Soc(R_R)-ann_R^l(M)$. Thanks for any cooperation!

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  • $\begingroup$ May I ask what's the motivation for your searching? $\endgroup$
    – Xam
    Commented May 6, 2017 at 21:22
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    $\begingroup$ @Xam I am zooming in the triangular matrix rings $[R, M, 0, S]$. Also, I have in mind the well-known fact that idempotents of the quotient $R/Soc(R_R)$ lift modulo $Soc(R_R)$. $\endgroup$
    – karparvar
    Commented May 7, 2017 at 14:01

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