Ring Homomorphism Preserves Negatives

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

Then:
 * $\forall x \in R_1: \map \phi {-x} = -\paren {\map \phi x}$

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

The result follows from Group Homomorphism Preserves Inverses.