Quotient Epimorphism is Epimorphism/Ring

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring whose zero is $$0_R$$ and whose unity is $$1_R$$.

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

Let $$\left({R / J, +, \circ}\right)$$ be the quotient ring defined by $$J$$.

Then the mapping $$\phi: R \to R / J$$ given by $$x \in R: \phi \left({x}\right) = x + J$$ is a ring epimorphism whose kernel is $$J$$.

This ring epimorphism is called the natural epimorphism from $$\left({R, +, \circ}\right)$$ onto $$\left({R / J, +, \circ}\right)$$.

Proof

 * The fact that $$\phi$$ is a homomorphism can be verified easily.


 * $$\phi$$ is surjective because $$\forall x + J \in R / J: x + J = \phi \left({x}\right)$$.

Therefore $$\phi$$ is an epimorphism.


 * Let $$x \in \ker \left({\phi}\right)$$. Then:

$$ $$ $$ $$

Thus $$\ker \left({\phi}\right) = J$$.