Ring Homomorphism Preserves Subrings

Theorem
Let $$\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$$ be a ring homomorphism.

If $$S$$ is a subring of $$R_1$$, then $$\phi \left({S}\right)$$ is a subring of $$R_2$$.

Proof 1

 * Since $$S \ne \varnothing$$, $$\phi \left({S}\right) \ne \varnothing$$.


 * From Group Homomorphism Preserves Subgroups, $$\left({\phi \left({S}\right), +_2}\right)$$ is a subgroup of $$\left({R_2, +_2}\right)$$.


 * From Homomorphism Preserves Subsemigroups, $$\left({\phi \left({S}\right), \circ_2}\right)$$ is a subsemigroup of $$\left({R_2, \circ_2}\right)$$.

Thus, as $$\left({R_2, +_2}\right)$$ is a group and $$\left({R_2, \circ_2}\right)$$ is a semigroup, the result follows.

Proof 2
From Morphism Property Preserves Closure, $$\phi \left({R_1}\right)$$ is a closed algebraic structure.

From Epimorphism from Ring, $$\phi \left({S}\right)$$ is a ring.

Hence the result, from the definition of subring.