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 $$\left({R_2, +_2, \circ_2}\right)$$.