Rings

Overview

Axioms

A ring {% R %} is a set with two binary operations defined. Addition is labeled {% a+b %} and multiplication is labeled {% ab %}. The two operations satisfy the following axioms.

1. {% a+b = b+a %}
2. {% (a+b) + c = a + (b+c) %}
3. There exists an element, labeled {% 0 %} such that {% a+0 = a %}. It is called the additive identity.
4. For each element {% a %}, there is another element in {% R %} labeled {% -a %} such that {% a + (-a) =0 %}
5. {% a(bc) = (ab)c %}
6. {% a(b+c) = ab + bc %} and also that {% (a+b)c = ac + bc %}