Socle ($\mathrm{Soc}(M)$)

The socle of a module is the sum of all its simple submodules, equivalently its largest semisimple module.

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 $J(A{)}^i$.