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
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Theorem $44 \ \text{(iii)}$