Universal Property of Quotient Ring

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Theorem

Let $R,S$ be commutative rings.

Let $I\trianglelefteq R$ be an ideal.

Let $\pi : R\to R/I$ be the projection.

Let $f:R\to S$ be a ring homomorphism with $f(I)=0$.


Then there exists a unique ring homomorphism $\overline f:R/I \to S$ such that $f = \overline f \circ \pi$.

$\xymatrix{ R \ar[d]_\pi \ar[r]^{\forall f} & S\\ R/I \ar[ru]_{\exists ! \bar f} }$


Proof


Also see