As the title says, I am interested on the socle of a quotient module. Let $R$ be a ring and $M,N$ two left $R$-modules. If $f: M \rightarrow N$ then $f(socle(M)) \subseteq socle(N)$. In particular, if $N\leq M$ and $\overline{\cdot}:M\rightarrow M/N$ is the canonical projection, then $\overline{socle(M)}\subseteq socle(M/N)$. Are there, in general, (non-trivial) circumstances that would guarantee equality? Under what conditions is $socle(M/N) \cong socle(M)/socle(N)$ true (if at all)?
If it helps, my rings are commutative and artinian.
This answer might be a kind of trivial. But I post anyway because I'm not sure what you mean by non-trivial, mathematically.
Claim. If $N$ is a direct summand of $M$ then $\operatorname{soc}(M/N) = \overline{\operatorname{soc}(M)}$.
Proof. Let $L$ be a complement of $N$ in $M$. Then $$ \overline{\operatorname{soc}(M)} = \overline{\operatorname{soc}(N \oplus L)} = \overline{\operatorname{soc}(N) \oplus \operatorname{soc}(L)} \cong \operatorname{soc}(L) \cong \operatorname{soc}(M/N). $$