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.ca, b, cR

    (a + b).c = a.c + b.ca, b, cR

    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 ...

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