Principal ideal

In mathematics, specifically ring theory, a principal ideal is an ideal formula_1 in a ring formula_2 that is generated by a single element formula_3 of formula_2 through multiplication by every element of formula_5 The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset formula_6 generated by a single element formula_7 which is to say the set of all elements less than or equal to formula_8 in formula_9
The remainder of this article addresses the ring-theoretic concept.
Definitions.
While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition.
If formula_2 is a commutative ring with identity, then the above three notions are all the same.
In that case, it is common to write the ideal generated by formula_3 as formula_25 or formula_26
Examples of non-principal ideal.
Not all ideals are principal.
For example, consider the commutative ring formula_27 of all polynomials in two variables formula_8 and formula_29 with complex coefficients. The ideal formula_30 generated by formula_8 and formula_29 which consists of all the polynomials in formula_27 that have zero for the constant term, is not principal. To see this, suppose that formula_34 were a generator for formula_35 Then formula_8 and formula_37 would both be divisible by formula_38 which is impossible unless formula_34 is a nonzero constant.
But zero is the only constant in formula_40 so we have a contradiction.
In the ring formula_41 the numbers where formula_42 is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider formula_43 and formula_44 These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are formula_45 and formula_46 they are not associates.
Related definitions.
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.
Examples of principal ideal.
The principal ideals in formula_47 are of the form formula_48 In fact, formula_47 is a principal ideal domain, which can be shown as follows. Suppose formula_50 where formula_51 and consider the surjective homomorphisms formula_52 Since formula_53 is finite, for sufficiently large formula_54 we have formula_55 Thus formula_56 which implies formula_1 is always finitely generated. Since the ideal formula_58 generated by any integers formula_3 and formula_60 is exactly formula_61 by induction on the number of generators it follows that formula_1 is principal.
However, all rings have principal ideals, namely, any ideal generated by exactly one element. For example, the ideal formula_63 is a principal ideal of formula_64 and formula_65 is a principal ideal of formula_66 In fact, formula_67 and formula_68 are principal ideals of any ring formula_5
Properties.
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 formula_70 to be any generator of the ideal formula_71
For a Dedekind domain formula_72 we may also ask, given a non-principal ideal formula_1 of formula_72 whether there is some extension formula_75 of formula_2 such that the ideal of formula_75 generated by formula_1 is principal (said more loosely, formula_1 "becomes principal" in formula_75).
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 formula_2 (i.e. the ring of integers of some number field) is contained in a larger integer ring formula_75 which has the property that "every" ideal of formula_2 becomes a principal ideal of formula_84 In this theorem we may take formula_75 to be the ring of integers of the Hilbert class field of formula_2; that is, the maximal unramified abelian extension (that is, Galois extension whose Galois group is abelian) of the fraction field of formula_72 and this is uniquely determined by formula_5
Krull's principal ideal theorem states that if formula_2 is a Noetherian ring and formula_1 is a principal, proper ideal of formula_72 then formula_1 has height at most one.