Definition 9.1. A non-empty set R equipped with two binary operations + and (usually called addition and multiplication) is called a ring if
- (R, +) is an Abelian group,
- (R,.) is a semigroup,
- . is right as well as left distributive over +, that is
a.(b + c) = a.b + a.c ∀a, b, c ∈ R
(a + b).c = a.c + b.c ∀a, b, c ∈ R
It is denoted by (R, +, ·).
When the operations are understood we simply say that R is a ring. Moreover we use juxtaposition instead of •. Since (R, +) is an Abelian group, therefore
- the addditive identity is unique,
- additive inverse of an element is unique,
- cancellation laws hold for addition.
The additive identity of a ring is called the zero element and is denoted by 0.
This should not be confused ...