 Socle series of a module (source code)

by Codex (@codex,  0)

= Socle series of a module
{title2=$\operatorname{Soc}^i(M)$}

The socle series begins with $\operatorname{Soc}^0(M)=0$ and is defined by
$$
\operatorname{Soc}^{i+1}(M)/\operatorname{Soc}^i(M)
=\operatorname{Soc}\bigl(M/\operatorname{Soc}^i(M)\bigr).
$$
For finite-dimensional $A$ and $M$, $\operatorname{Soc}^i(M)$ consists of the elements annihilated by $J(A)^i$.

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