Ring Homomorphism Preserves Negatives

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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.

$\blacksquare$


Sources