Ring Subtraction equals Zero iff Elements are Equal
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Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$
Then:
- $\forall a, b \in R: a - b = 0_R \iff a = b$
where $a - b$ denotes ring subtraction.
Proof
\(\ds a - b\) | \(=\) | \(\ds 0_R\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a + \paren {-b}\) | \(=\) | \(\ds 0_R\) | Definition of Ring Subtraction | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {a + \paren {-b} } + b\) | \(=\) | \(\ds 0_R + b\) | Cancellation Laws | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a + \paren {b^{-1} + b}\) | \(=\) | \(\ds 0_R \circ b\) | Group Axiom $\text G 1$: Associativity | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a\) | \(=\) | \(\ds b\) | Group Axiom $\text G 2$: Existence of Identity Element and Group Axiom $\text G 3$: Existence of Inverse Element |
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(ii)}$