Homomorphism of Ring Subtraction

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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:

$\forall a, b \in R_1: \map \phi {a -_1 b} = \map \phi a -_2 \map \phi b$

where $a -_1 b$ denotes subtraction of $b$ from $a$.


Proof

\(\ds \map \phi {a -_1 b}\) \(=\) \(\ds \map \phi {a +_1 \paren {-b} }\) Definition of Ring Subtraction
\(\ds \) \(=\) \(\ds \map \phi a +_2 \map \phi {-b}\) Definition of Ring Homomorphism
\(\ds \) \(=\) \(\ds \map \phi a +_2 \paren {-\map \phi b}\) Ring Homomorphism Preserves Negatives
\(\ds \) \(=\) \(\ds \map \phi a -_2 \map \phi b\) Definition of Ring Subtraction

$\blacksquare$


Sources