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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.
- 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 ideal
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 ideal
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 .
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.
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 .