Universal Property of Quotient Ring

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Let $R, S$ be commutative rings.

Let $I \trianglelefteq R$ be an ideal of $R$.

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

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

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

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


Define $\overline f: R / I \to S$ by:

$\forall r \in R: \map {\overline f} {r + I} = \map f r$

Since $f$ is a ring homomorphism, $f$ is well-defined.

Suppose for some $r_1, r_2 \in R$ that:

$r_1 + I = r_2 + I$

Since $I$ is an ideal, it contains the zero $0_R$ of $R$.

Then for some $i \in I$, we have that:

\(\ds r_1 + 0_R\) \(=\) \(\ds r_1\) Definition of Ring Zero
\(\ds \) \(=\) \(\ds r_2 + i\) Definition of Set Equality


\(\ds \map {\overline f} {r_1 + I}\) \(=\) \(\ds \map f {r_1}\) Definition of $\overline f$
\(\ds \) \(=\) \(\ds \map f {r_2 + i}\) Since $r_1=r_2+i$
\(\ds \) \(=\) \(\ds \map f {r_2} + \map f i\) Definition of Ring Homomorphism
\(\ds \) \(=\) \(\ds \map f {r_2} + 0_S\) Since $f(I)=\{0\}$
\(\ds \) \(=\) \(\ds \map f {r_2}\) Definition of Ring Zero
\(\ds \) \(=\) \(\ds \map {\overline f} {r_2 + I}\) Definition of $\overline f$

so $\overline f$ is seen to be well-defined.

Next, for all $r_1, r_2 \in R$:

\(\ds \map {\overline f} {r_1 + I + r_2 + I}\) \(=\) \(\ds \map {\overline f} {r_1 + r_2 + I}\) Quotient Ring Addition is Well-Defined
\(\ds \) \(=\) \(\ds \map f {r_1 + r_2}\) Definition of $\overline f$
\(\ds \) \(=\) \(\ds \map f {r_1} + \map f {r_2}\) Definition of Ring Homomorphism
\(\ds \) \(=\) \(\ds \map {\overline f} {r_1 + I} + \map {\overline f} {r_2 + I}\) Definition of $\overline f$

Therefore $\overline f$ is a homomorphism.

It remains to demonstrate uniqueness.

Suppose there were another homomorphism $g: R / I \to S$ with $f = g \circ \pi$.

Then we would require that, for all $r\in R$:

\(\ds \map g {r + I}\) \(=\) \(\ds \map f r\) Definition of Commutative Diagram
\(\ds \) \(=\) \(\ds \map {\overline f} {r + I}\) Definition of $\overline f$

That is:

$g = \overline f$

and so $\overline f$ is unique.


Also see