Composition of Ring Epimorphisms is Ring Epimorphism

Theorem
Let:
 * $\left({R_1, +_1, \circ_1}\right)$
 * $\left({R_2, +_2, \circ_2}\right)$
 * $\left({R_3, +_3, \circ_3}\right)$

be rings.

Let:
 * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$
 * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$

be (ring) epimorphisms.

Then the composite of $\phi$ and $\psi$ is also a (ring) epimorphism.

Proof
A ring epimorphism is a ring homomorphism which is also a surection.

From Composition of Ring Homomorphisms is Ring Homomorphism, $\psi \circ \phi$ is a ring homomorphism.

From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surection.