Rings

Introduction

A ring is a group R with an additional associative operation * and distributive property.

Axioms

A ring is a set R together with two binary operations + and * such that the following axioms hold.

Additive Group Axioms

∀ x, y ∈ R, x + y ∈ R ∀ x, y, z ∈ R, x + (y + z) = (x + y) + z ∀ x, y ∈ R, x + y = y + x ∃ 0 ∈ R, ∀ x ∈ R, x + 0 = x ∀ x ∈ R, ∃ -x ∈ R such that x + -x = 0

Multiplicative Axioms

∀ x, y ∈ R, x * y ∈ R ∀ x, y, z ∈ R, x * (y * z) = (x * y) * z ∃ 1 ∈ R, ∀ x ∈ R, x * 1 = x

Distributive Axioms

∀ x, y, z ∈ R, x * (y + z) = x * y + x * z ∀ x, y, z ∈ R, (x + y) * z = x * z + y * z