Socles and factors

Question

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle.

Let $f: M \longrightarrow N$ be an epimorphism and let $\iota:M' \longrightarrow M$ be the inclusion map, and assume that $f \circ \iota :M' \longrightarrow N$ is still an epimorphism.

Question:

If the simple socle of $N$ is a summand of the socle of $M$, will it also be a summand of the socle of $M'$?

To be precise: if the restriction of the map $f$ to $\operatorname{Soc}(M)$, $$f_|:\operatorname{Soc}(M) \longrightarrow \operatorname{Soc}(N),$$ is an epimorphism (a split epic), will the restriction of $f \circ \iota$ to $\operatorname{Soc}(M')$, $$(f \circ \iota)_|:\operatorname{Soc}(M') \longrightarrow \operatorname{Soc}(N),$$ be a split epic as well?

• Intuitively I feel that the answer has to affirmative, but I have not been able to prove this, nor find a counterexample. Does anyone know if this is true? Any help would be immensely appreciated.

Answer 1

No. For example, suppose $N=S$ is simple, so that $\operatorname{soc}N=S$, $M'$ is a non-split extension of $S$ by another simple module $T$, so that $\operatorname{soc}M'=T$ and there is an epimorphism $\alpha:M'\to N$ which is zero on $\operatorname{soc}M'$, and that $M=M'\oplus S$.

Then the map $\begin{pmatrix}\alpha&\operatorname{id}_S\end{pmatrix}:M'\oplus S\to S$ induces a split epimorphism $\operatorname{soc}M=T\oplus S\to\operatorname{soc}N=S$.