Quotient Epimorphism is Epimorphism/Ring

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

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

Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$.

Let $\phi: R \to R / J$ be the quotient (ring) epimorphism from $R$ to $R / J$:
 * $x \in R: \map \phi x = x + J$

Then $\phi$ is a ring epimorphism whose kernel is $J$.

Proof
Let $x, y \in R$.

Then:

and:

Thus $\phi$ is a homomorphism.

$\phi$ is surjective because:
 * $\forall x + J \in R / J: x + J = \map \phi x$

Therefore $\phi$ is an epimorphism.

Let $x \in \map \ker \phi$.

Then:

Thus:
 * $\map \ker \phi = J$