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In lectures about ring theory my professor says that a map factorizes over a ring and then draws commutative diagrams. I have never heard this expression before.

What exactly does it mean for a map to factorize? Is it equivalent to the diagram commuting or is it stronger? Why this name?

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When you have a commutative diagram (triangle) that captures $h = g \circ f$, it has the same appearance as factoring an integer such as $12 = 3 \cdot 4$, hence the terminology: the integer $12$ has been decomposed into constituent components, and the same is true for $h$. The most fundamental appearance of this that you should master is the first isomorphism theorem: a surjective ring hom $h: R \to T$ factors through a quotient of the kernel, the factors being the quotient map $R \to R/\mathrm{ker}(h)$ and the resulting induced map $R/\mathrm{ker}(h) \to T$ (an iso!).

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