Ring Homomorphism whose Kernel contains Ideal

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Theorem

Let $R$ be a ring.

Let $J$ be an ideal of $R$.

Let $\nu: R \to R / J$ be the quotient epimorphism.


Let $\phi: R \to S$ be a ring homomorphism such that:

$J \subseteq \ker \left({\phi}\right)$

where $\ker \left({\phi}\right)$ is the kernel of $\phi$.


Then there exists a unique ring homomorphism $\psi: R / J \to S$ such that:

$\phi = \psi \circ \nu$

where $\circ$ denotes composition of mappings.

CommDiagRingHomKerIdeal.png

Also:

$\ker \left({\psi}\right) = \ker \left({\phi}\right) / J$


Proof

Suppose $\phi = \psi \circ \nu$.

Let $J + x$ be an arbitrary element of $R / J$.

Then:

$(1) \quad \psi \left({J + x}\right) = \psi \circ \nu \left({x}\right) = \phi \left({x}\right)$

So there is only one possible way to define $\psi$.

Now suppose $J + x = J + x'$.

Then $x + \left({-x'}\right) \in J$.

So $x + \left({-x'}\right) \in \ker \left({\phi}\right)$ as $J \subseteq \ker \left({\phi}\right)$.

That is, $\phi \left({x + \left({-x'}\right)}\right) = 0_S$.

So $\phi \left({x}\right) = \phi \left({x'}\right)$ and so $\psi$ as defined in $(1)$ is well-defined.


Now suppose $J + x, J + y \in R / J$.

We have:

\(\displaystyle \psi \left({\left({J + x}\right) + \left({J + y}\right)}\right)\) \(=\) \(\displaystyle \psi \left({J + \left({x + y}\right)}\right)\) Definition of Quotient Ring
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({x + y}\right)\) Definition of $\psi$ in $(1)$ above
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({x}\right) + \phi \left({y}\right)\) Definition of Ring Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle \psi \left({J + x}\right) + \psi \left({J + y}\right)\)

So $\psi$ preserves ring addition.


Then:

\(\displaystyle \psi \left({\left({J + x}\right) \left({J + y}\right)}\right)\) \(=\) \(\displaystyle \psi \left({J + \left({x y}\right)}\right)\) Definition of Quotient Ring
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({x y}\right)\) Definition of $\psi$ in $(1)$ above
\(\displaystyle \) \(=\) \(\displaystyle \phi \left({x}\right) \phi \left({y}\right)\) Definition of Ring Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle \psi \left({J + x}\right) \psi \left({J + y}\right)\)

So $\psi$ preserves ring product, and so $\psi$ is a ring homomorphism.


Finally:

\(\displaystyle \psi \left({J + x}\right)\) \(=\) \(\displaystyle 0_S\)
\(\displaystyle \iff \ \ \) \(\displaystyle \phi \left({x}\right)\) \(=\) \(\displaystyle 0_S\)
\(\displaystyle \iff \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \ker \left({\phi}\right)\)

So:

$\ker \left({\psi}\right) = \ker \left({\phi}\right) / J$

$\blacksquare$


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