Proofs of elementary ring properties
The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Basics[edit]
Multiplication by zero[edit]
Theorem:
By subtracting (i.e. adding the additive inverse of) on both sides of the equation, we get the desired result. The proof that is similar.
Unique identity element per binary operation[edit]
Theorem: The identity element e for a binary operation (addition or multiplication) of a ring is unique.
Unique additive inverse element[edit]
Theorem: - a as the additive inverse element for a is unique.
Unique multiplicative inverse element[edit]
Theorem: a−1 as the multiplicative inverse element for a is unique.
Zero ring[edit]
Theorem: A ring is the zero ring (that is, consists of precisely one element) if and only if .
Multiplication by negative one[edit]
Theorem:
Therefore .
Multiplication by additive inverse[edit]
Theorem:
To prove that the first expression equals the second one,
To prove that the first expression equals the third one,
A pseudo-ring does not necessarily have a multiplicative identity element. To prove that the first expression equals the third one without assuming the existence of a multiplicative identity, we show that is indeed the inverse of by showing that adding them up results in the additive identity element,
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