Theorem

$\mathcal Z(R_R)\subseteq \ell.ann(soc(R_R))$, and they are equal when $R$ is right Artinian.

For any ring $R$, the right singular ideal $\mathcal Z(R_R)\subseteq \ell.ann(soc(R_R))$. When $R$ is right Artinian, equality holds.

Reference(s)

  • T. Lam. Lectures on modules and rings. (2012) @ Proposition 7.13 p 251