I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is the proof of Lemma 2.2

$A$ is assumed to be an Artin algebra and mod(A) the category of finitely generated $A$ - modules. By $S \rightarrow I_s$ one means the injective hull of a simple module $S$. $P_s \rightarrow S$ denoted the projective cover of $S$.

The sentence underlined in pink confuses me the most. How does it follow form $g \neq 0$ that $S_r=S$? Any help is highly appreciated!