Principal ideal

The principal ideal is a term used in ring theory, a branch of algebra. It represents a generalization of the familiar from school mathematics subsets of the integers that are multiples of a number. Examples of subsets are even numbers or multiples of the number 3

Definition

A principal ideal of the ring is an element produced from a single ideal

Properties

The complex products

And

Valid for the generated by

• Principal left ideal:
• Principal right ideal:
• ( two-sided ) principal ideal:

If the ring has an identity element 1, it follows for the

• Principal left ideal:
• Principal right ideal:
• ( two-sided ) principal ideal:

Comments

• In commutative rings, however, do not match all three types of principal ideals, in general.
• Not every ideal of a ring must be a principal ideal.

Example

As an example we consider the commutative ring of all polynomials in two indeterminates over a field K. The ideal of X and Y generated by the two polynomials consists of all polynomials whose constant term is equal to 0. This ideal is not a principal ideal, because if a polynomial P is a generator of then takes P is a divisor of both X and Y be as applicable only to the constant polynomials different from 0. However, these are not included in.

Related term

An integral domain in which every ideal is a principal ideal is called principal ideal ring.