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 $\text{mod}(A)$ and let $\iota_S: S \rightarrow I(S)$ be the injective envelope of $S$. Then the natural epimorphism $p: I(S) \rightarrow I(S)/\text{Im}(\iota_S)$ is irreducible.
Unfortunately I have no idea how to prove this. Can anybody help?
Suppose not. Then $p$ factors as $$I(S)\xrightarrow{s}X\xrightarrow{t}I(S)/\operatorname{Im}(\iota_S),$$ where $s$ is not a split monomorphism.
Since $I(S)$ is injective, $s$ can not be a monomorphism, or it would be a split monomorphism. Since $\operatorname{Im}(\iota_S)$ is the unique simple submodule of $I(S)$, it must be contained in the kernel of $s$, and so $s$ factors through $p$: say $s=rp$ for some $r:I(S)/\operatorname{Im}(\iota_S)\to X$. So $p=ts=trp$, and so $tr$ is the identity map of $I(S)/\operatorname{Im}(\iota_S)$, since $p$ is surjective.
So $t$ is a split epimorphism.