Ring Homomorphism Preserves Subrings/Corollary

Corollary to Ring Homomorphism Preserves Subrings
Let $\left({R_1, +_1, \circ_1}\right)$ and $\left({R_2, +_2, \circ_2}\right)$ be rings.

The image of a ring homomorphism $\phi: R_1 \to R_2$ is a subring of $R_2$.

Proof
From Null Ring and Ring Itself Subrings, $R_1$ is a subring of itself.

The result then follows from Ring Homomorphism Preserves Subrings.