Ring Epimorphism Preserves Subrings

Theorem

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism.

Let $S$ be a subring of $R_1$.

Then $\phi \sqbrk S$ is a subring of $R_2$.

Proof

A direct application of Ring Homomorphism Preserves Subrings.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.6: \ 2^\circ$