N \subseteq M

for which

M / N

is a semisimple module. Consequently, if

M = M_{0} \supseteq M_{1} \supseteq M_{2} \supseteq \dots

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 \dots

has semisimple successive quotients, induction in

M / N_{i}

gives

N_{i} \subseteq 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)^{i} M, Soc^{i} (M) = {x \in M : J (A)^{i} x = 0} .

Thus

J^{r} (M) = 0

exactly when

J (A)^{r}

annihilates all of

M

, which is exactly when

Soc^{r} (M) = M

. The two least terminating indices therefore coincide:

m = n = the Loewy length of M .

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