Clarification about stable module category

Question

I'm trying to read about stable module categories, and in the definition they say, "$f \sim g$ if $f − g$ factors through a projective module", where $f$ and $g$ are module homomorphism. I'm just not sure what it means for maps to "factor through" modules. This comes from this Wikipedia article: https://en.wikipedia.org/wiki/Stable_module_category

Answer 1

Let $f\colon X \to Y$ be a morphism in a category and let $Z$ be an object. We say that $f$ factors over $Z$ if there are maps $g\colon X \to Z$ and $h\colon Z \to Y$ such that $f = h \circ g$.

In your specific case, this is to say that $f \sim g\colon M \to N$ if there exists some projective module $P$ and maps $\alpha\colon M \to P$, $\beta\colon P \to N$ such that $f - g = \beta \circ \alpha$.