Epimorphism Preserves Rings

Theorem
Let $\left({R_1, +_1, \circ_1}\right)$ be a ring, and $\left({R_2, +_2, \circ_2}\right)$ be a closed 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 Groups, we have that if $\left({R_1, +_1}\right)$ is a group then so is $\left({R_2, +_2}\right)$.


 * From Epimorphism Preserves Semigroups, we have that if $\left({R_1, \circ_1}\right)$ is a semigroup then so is $\left({R_2, \circ_2}\right)$.


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

So it follows from the definition of a ring that if $\left({R_1, +_1, \circ_1}\right)$ is a ring then so is $\left({R_2, +_2, \circ_2}\right)$.