The <radical of a module> is the smallest submodule $N\subseteq M$ for which $M/N$ is a <semisimple module>. Consequently, if
$$
M=M_0\supseteq M_1\supseteq M_2\supseteq\cdots
$$
has semisimple successive quotients, then $J(M_i)\subseteq M_{i+1}$. Induction gives
$$
J^i(M)\subseteq M_i,
$$
so the <radical series of a module> descends at least as fast as every such series.

Dually, the <socle> is the largest semisimple submodule. If
$$
0=N_0\subseteq N_1\subseteq N_2\subseteq\cdots
$$
has semisimple successive quotients, induction in $M/N_i$ gives
$$
N_i\subseteq\operatorname{Soc}^i(M),
$$
so the <socle series of a module> ascends at least as fast as every such series.

Both series terminate because $M$ has finite <composition length>. More precisely,
$$
J^i(M)=J(A)^iM,
\qquad
\operatorname{Soc}^i(M)=\{x\in M:J(A)^ix=0\}.
$$
Thus $J^r(M)=0$ exactly when $J(A)^r$ annihilates all of $M$, which is exactly when $\operatorname{Soc}^r(M)=M$. The two least terminating indices therefore coincide:
$$
\boxed{m=n=\text{the Loewy length of }M.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 1 / c / Solution

{title2=$\ell(M)$}

The composition length of a finite-length module or group is the number of simple factors in a <composition series>. The <Jordan–Hölder theorem> makes this independent of the chosen composition series.

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 Incoming links: Composition length