Ring Homomorphism of Addition is Group Homomorphism

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

Then $\phi: \left({R_1, +_1}\right) \to \left({R_2, +_2}\right)$ is a group homomorphism.

Proof
From the definition of a ring, both $\left({R_1, +_1}\right)$ and $\left({R_2, +_2}\right)$ are abelian groups.

The result follows from the definitions of ring homomorphism and group homomorphism.