The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Contents
Multiplication by zero
Theorem: 0 ⋅ a = a ⋅ 0 = 0
Zero ring
Theorem: A ring (R, +, ⋅) is the zero ring (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)
References
Proofs of elementary ring properties Wikipedia(Text) CC BY-SA