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

$\blacksquare$


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