Principal ideal

In mathematics, specifically ring theory, a principal ideal is an ideal

that is generated by a single element

The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset

generated by a single element

which is to say the set of all elements less than or equal to

The remainder of this article addresses the ring-theoretic concept.

While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under addition.

is a commutative ring, then the above three notions are all the same.

In that case, it is common to write the ideal generated by

A ring in which every ideal is principal is called principal, or a principal ideal ring.

A principal ideal domain (PID) is an integral domain in which every ideal is principal.

Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.

is a principal ideal domain, which can be shown as follows.

and consider the surjective homomorphisms

is finite, for sufficiently large

is always finitely generated.

generated by any integers

by induction on the number of generators it follows that

Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal.

More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication.

In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define

This question arose in connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in number theory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others.

The principal ideal theorem of class field theory states that every integer ring

(i.e. the ring of integers of some number field) is contained in a larger integer ring

to be the ring of integers of the Hilbert class field of

; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of

and this is uniquely determined by

Krull's principal ideal theorem states that if

is a Noetherian ring and

is a principal, proper ideal of