Let $R$ be a ring and $M$ a $R$-module. The we define $\text{rad}^n(M) := \text{rad}(\text{rad}^{n-1}(M))$ and $\text{soc}^n(M)$ as the preimage in $M$ of the socle of $M/\text{soc}^{n-1}(M)$. Now I have no idea how to prove the following two statements:
(i) $\text{soc}^n(M) = M$ iff $\text{rad}^n(M) = 0$,
(ii) for a morhpism of modules $f: M \rightarrow N$ we have $f(\text{soc}^n(M)) \subset \text{soc}^n(N)$.
My biggest problem is the power n, I think you cannot show the equality straightforward. What's the trick here?
A hint or two would be great! Thank you in advance!