Epimorphism Preserves Rings

Theorem
Let $$\left({R_1, +_1, \circ_1}\right)$$ be a ring, and $$\left({R_2, +_2, \circ_2}\right)$$ be an algebraic structure.

Let $$\phi: R_1 \to R_2$$ be an epimorphism.

Then $$\left({R_2, +_2, \circ_2}\right)$$ is a ring.

Proof
From Epimorphism Preserves Distributivity, we have that $$\circ_2$$ distributes over $$+_2$$ if $$\circ_1$$ distributes over $$+_1$$.

So by Homomorphism Preserves Subsemigroups and Homomorphism Preserves Subgroups etc., it follows that $$\left({R_2, +_2, \circ_2}\right)$$ is a ring.