Proofs of elementary ring properties

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

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: (&minus;1)a = &minus;a

Multiplication by additive inverse
Theorem 3: (&minus;a) ⋅ b = a ⋅ (&minus;b) = &minus;(ab)