Socle series and radical series have same length

Question

I have been given a module $V$ of finite length $l(V)=n$. If I am not wrong if follows from Jordan- Hölder theorem that the socle series and radical series have also finite length since they are filtrations (I proved that in each step we gain at least one dimension), but now I am asked to prove that both series have same finite length.

If $V$ is semisimple it follows that the length is 1 in both cases...could it be some kind of induction valid?

Moreover, I have tried to relate both series, but I would appreciate any hint since I am a little bit confused. I have been given the following hint: "prove that the composition $rad(V) \longrightarrow V \longrightarrow V/Soc^{n-1}(V)$ is zero", but I don't know how to use it...

Thanks in advance!

Answer 1

Let $A$ be the underlying ring.

Note that $Soc(V)= (0 :_V Rad (A))$.

And due to the finiteness condition, $Rad (V)= (Rad(A))V$

(in general $Rad (V) \supseteq (Rad(A))V$).

So, $$Rad^k(V)= (Rad(A))^kV$$ and $$Soc^k(V)= (0 :_V (Rad (A))^k)$$ So if $k_1$ is the least integer such that $(Rad(A))^{k_1}V=0$ then $(0 :_V (Rad (A))^{k_1})=V$ and $k_2$ is the least integer such that $(0 :_V (Rad (A))^{k_2})=V$ then $(Rad(A))^{k_2}V=0$ implying $k_1=k_2$