# Ring Subtraction equals Zero iff Elements are Equal

## 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

 $\displaystyle a - b$ $=$ $\displaystyle 0_R$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a + \paren {-b}$ $=$ $\displaystyle 0_R$ Definition of Ring Subtraction $\displaystyle \leadstoandfrom \ \$ $\displaystyle \paren {a + \paren {-b} } + b$ $=$ $\displaystyle 0_R + b$ Cancellation Laws $\displaystyle \leadstoandfrom \ \$ $\displaystyle a + \paren {b^{-1} + b}$ $=$ $\displaystyle 0_R \circ b$ Group Axiom $\text G 1$: Associativity $\displaystyle \leadstoandfrom \ \$ $\displaystyle a$ $=$ $\displaystyle b$ Group Axiom $\text G 2$: Existence of Identity Element and Group Axiom $\text G 3$: Existence of Inverse Element

$\blacksquare$