# 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.

$\blacksquare$