Ring Homomorphism Preserves Negatives

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

Then:
 * $\forall x \in R_1: \phi \left({-x}\right) = -\phi \left({x}\right)$

Proof
We have that Ring Homomorphism of Addition is Group Homomorphism.

The result follows from Group Homomorphism Preserves Inverses.