Ring Homomorphism Preserves Subrings/Proof 2

Theorem

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

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

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

Proof

From Morphism Property Preserves Closure, $\phi \sqbrk {R_1}$ is a closed algebraic structure.

From Epimorphism Preserves Rings, $\phi \sqbrk S$ is a ring.

Hence the result, from the definition of subring.

$\blacksquare$