Algebra I: A Basic Course in Abstract Algebra

Chapter 9

Rings

9.1 Ring

Definition 9.1.    A non-empty set R equipped with two binary operations + and (usually called addition and multiplication) is called a ring if

1. (R, +) is an Abelian group,
2. (R,.) is a semigroup,
3. . 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

1. the addditive identity is unique,
2. additive inverse of an element is unique,
3. 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 ...