Proofs of elementary ring properties

The following proofs of elementary ring properties use only the axioms that define a mathematical ring:

Contents

Basics

Multiplication by zero

Theorem: 0 ⋅ a = a ⋅ 0 = 0

Trivial ring

Theorem: A ring (R, +, ⋅) is trivial (that is, consists of precisely one element) if and only if 0 = 1.

Multiplication by negative one

Theorem: (−1)a = −a

Multiplication by additive inverse

Theorem 3: (−a) ⋅ b = a ⋅ (−b) = −(ab)