Ring Homomorphism of Addition is Group Homomorphism

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

Then $\phi: \struct {R_1, +_1} \to \struct {R_2, +_2}$ is a group homomorphism.

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

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