Principal ideal

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.

DefinitionsEdit

  • 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 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.[citation needed]

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 of non-principal idealEdit

Not all ideals are principal. For example, consider the commutative ring   of all polynomials in two variables   and   with complex coefficients. The ideal   generated by   and   which consists of all the polynomials in   that have zero for the constant term, is not principal. To see this, suppose that   were a generator for   Then   and   would both be divisible by   which is impossible unless   is a nonzero constant. But zero is the only constant in   so we have a contradiction.

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 definitionsEdit

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 idealEdit

The principal ideals in   are of the form   In fact,   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.

However, all rings have principal ideals, namely, any ideal generated by exactly one element. For example, the ideal   is a principal ideal of   and   is a principal ideal of   In fact,   and   are principal ideals of any ring  

PropertiesEdit

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   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 alsoEdit

ReferencesEdit

  • Gallian, Joseph A. (2017). Contemporary Abstract Algebra (9th ed.). Cengage Learning. ISBN 978-1-305-65796-0.