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In the mathematical field of 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.

Contents

DefinitionsEdit

  • a left principal ideal of   is a subset of   of the form  ;
  • a right principal ideal of   is a subset of the form  ;
  • a two-sided principal ideal of   is a subset of all finite sums of elements of the form  , namely,  .

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   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  .

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  , but because the only units in the ring are   and  , they are not associates.

Examples of principal idealEdit

The principal ideals in   are of the form  . Actually, every ideal   in   is principal, which can be shown in the following way. Suppose   where  , then consider the surjective homomorphisms   Since   is finite, then for sufficiently large  ,   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  .

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

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.