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In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of 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 in
The remainder of this article addresses the ring-theoretic concept.
Definitions
edit- A left principal ideal of is a subset of given by for some element
- A right principal ideal of is a subset of given by for some element
- A two-sided principal ideal of is a subset of given by for some element namely, the set of all finite sums of elements of the form
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.[1]: 251–252
If 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 as or
Examples and non-examples
edit- The principal ideals in the (commutative) ring are In fact, every ideal of is principal (see § Related definitions).
- In any ring , the sets and are principal ideals.
- For any ring and element the ideals and are respectively left, right, and two-sided principal ideals, by definition. For example, is a principal ideal of
- In the commutative ring of complex polynomials in two variables, the set of polynomials that vanish everywhere on the set of points is a principal ideal because it can be written as (the set of polynomials divisible by ).
- In the same ring , the ideal generated by both and is not principal. (The ideal is the set of all polynomials with zero for the constant term.) To see this, suppose there was a generator for so Then contains both and so must divide both and Then must be a nonzero constant polynomial. This is a contradiction since but the only constant polynomial in is the zero polynomial.
- In the ring the numbers where is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider and These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are and they are not associates.
Related definitions
editA 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.
As an example, is a principal ideal domain, which can be shown as follows. Suppose where and consider the surjective homomorphisms Since is finite, for sufficiently large we have Thus which implies is always finitely generated. Since the ideal generated by any integers and is exactly by induction on the number of generators it follows that is principal.
Properties
editAny 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 to be any generator of the ideal
For a Dedekind domain we may also ask, given a non-principal ideal of whether there is some extension of such that the ideal of generated by is principal (said more loosely, becomes principal in ). 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 which has the property that every ideal of becomes a principal ideal of In this theorem we may take 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 then has height at most one.
See also
editReferences
edit- ^ Dummit, David S.; Foote, Richard M. (2003-07-14). Abstract Algebra (3rd ed.). New York: John Wiley & Sons. ISBN 0-471-43334-9.
- Gallian, Joseph A. (2017). Contemporary Abstract Algebra (9th ed.). Cengage Learning. ISBN 978-1-305-65796-0.