Commutative ring/Ideal/Introduction/Section


A subset of a commutative ring is called an ideal, if the following conditions are fulfilled:

  1. .
  2. For all , we have .
  3. For all and , we have .

The property can be replaced by the condition that is not empty. An ideal is a subgroup of the additive group of , which, moreover, is also closed under scalar multiplication.


For a family of element in a commutative ring , we denote by the ideal generated by these elements. It consists of all linear combinations

where

.


An ideal in a commutative ring of the form

is called a principal ideal.

The zero element forms, in every ring, the so-called zero ideal. We write this simply as . write. The and, moreover, every unit, generates as an ideal the total ring.


The unit ideal in a commutative ring

is the ring itself.

In a field, there exist exactly two ideals.


Let be a commutative ring. Then the following statements are equivalent.

  1. is a field.
  2. There exist exactly two ideals in .

If is a field, then there exists the zero ideal and the unit ideal, and these are different ideals. Let be an ideal in different from . Then contains some element , which is a unit. Therefore, and thus .

Suppose now that is a commutative ring with exactly two ideals. Then is not the zero ring. Let now be an element in different from . The principal ideal generated by , that is , is , and therefore it must be the other ideal, which is the unit ideal. In particular, this means . Hence, for some , so that is a unit.