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.

$\ds a - b$

$=$

$\ds 0_R$

$\ds \leadstoandfrom \ \ $

$\ds a + \paren {-b}$

$=$

$\ds 0_R$

Definition of Ring Subtraction

$\ds \leadstoandfrom \ \ $

$\ds \paren {a + \paren {-b} } + b$

$=$

$\ds 0_R + b$

$\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

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(ii)}$