Socle series of a module (${\mathrm{Soc}}^i(M)$)

The socle series begins with ${\mathrm{Soc}}^0(M)=0$ and is defined byFor finite-dimensional $A$ and $M$, ${\mathrm{Soc}}^i(M)$ consists of the elements annihilated by $J(A{)}^i$.