# Epimorphism Preserves Rings

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## Theorem

Let $\struct {R_1, +_1, \circ_1}$ be a ring, and $\struct {R_2, +_2, \circ_2}$ be a closed algebraic structure.

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

Then $\struct {R_2, +_2, \circ_2}$ is a ring.

## Proof

From Epimorphism Preserves Groups, we have that if $\struct {R_1, +_1}$ is a group then so is $\struct {R_2, +_2}$.

From Epimorphism Preserves Semigroups, we have that if $\struct {R_1, \circ_1}$ is a semigroup then so is $\struct {R_2, \circ_2}$.

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 $\struct {R_1, +_1, \circ_1}$ is a ring then so is $\struct {R_2, +_2, \circ_2}$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$: Theorem $22.1$