Proofs of elementary ring properties

The following proofs of elementary ring properties use only the axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s that define a mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

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)

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