Proofs of elementary ring properties

The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Basics.
Multiplication by zero.
Theorem: formula_1
Unique identity element per binary operation.
Theorem: The identity element "e" for a binary opertaion (addition or multiplication) of a ring is unique.
Unique additive inverse element.
Theorem: "- a" as the additive inverse element for a is unique.
Unique multiplicative inverse element.
Theorem: "a"-1 as the multiplicative inverse element for a is unique.
Zero ring.
Theorem: A ring formula_2 is the zero ring (that is, consists of precisely one element) if and only if formula_3.
Multiplication by negative one.
Theorem: formula_4
Multiplication by additive inverse.
Theorem: formula_5