I am reading the book Elements of the Representation Theory of Associative Algebras, volume 1, by Assem et al. On page 162, it is written $D(soc^{i}M)\cong DM/rad^{i}DM$, where $DM=\mathrm{Hom}(M,k)$.
We define $soc^{i}M$ inductively as follows: $soc^{0}M=0$ and, if $soc^{i}M$ is already defined and $p:M\rightarrow M/soc^{i}M$ denotes the canonical epimorphism, we set $soc^{i+1}M=p^{-1}(soc(M/soc^{i}M))$. Thus, by definition, $soc^{i+1}M\supset soc^{i}M$. Then we get a socle series: $0=soc^{0}M\subset socM \subset soc^{2}M \subset \cdots \subset M$.
I don't know how to prove the identity $D(soc^{i}M)\cong DM/rad^{i}DM$, for any integer $i\geq 0$. Any help will be greatly appreciated! Thank you very much. In particular, for $i=1$, $D(socM)\cong DM/radDM$.