An Exercise on projective covers and injective envelopes of simple modules over an Artin algebra

Question

Let $A$ be an artin algebra and let $\text{mod}(A)$ denote the category of finitely generated left $A$-modules.

Let $S$ be a simple module in $\text{mod}(A)$. Let $\iota_S: S \rightarrow I(S)$ and $\pi_S: P(S) \rightarrow S$ be its injective envelope and projective cover, respectively. If $f:M \rightarrow I(S)$ is a non-zero monomorphism in $\text{mod}(A)$, then there exists a morphism $g: P(S) \rightarrow M$ such that $fg \neq 0$.

My attempt:

If $f$ is injective then we have an exact sequence $0 \rightarrow M \xrightarrow{f} I(S) \xrightarrow{p} I(S)/\text{Im}(f) \rightarrow 0$. Hence if we show $p\iota_S \pi_S=0$ then there must exist a morphism $g: P(S) \rightarrow M$ with $fg=\iota_S \pi_S \neq 0$.

Now in order to show $p\iota_S \pi_S=0$ I think one needs to prove that $S$ is a submodule of $\text{Im}(f)$. That is the point where I have no idea how to continue. Does anybody have a clue?

Answer 1

$S$ is the unique simple submodule of $I(S)$ [how easy this is to see depends on the definition of "injective envelope" that you are using]. So every nonzero submodule of $I(S)$ contains $S$.