Principal ideal domain

In algebra, a branch of mathematics, one calls integrity rings as principal ideal rings or principal ideal areas if every ideal is a principal ideal. The most important examples of principal ideal rings are the ring of integers and polynomial rings in one variable over a field. The concept of principal ideal ring allows it to formulate statements about these two special cases uniformly. Examples of applications of the general theory, the Jordan normal form, the partial fraction expansion or the structure theory of finitely generated abelian groups.

Definition

An integral domain (that is a zero-divisor free commutative ring with ) is called a principal ideal ring if every ideal is a principal ideal, ie there is a such that.

Below is a principal ideal ring and its quotient field. In addition, there is a set that contains exactly one element to associate for each irreducible. In the case the amount of the (positive) primes such, in the case of a body, the amount of irreducible polynomials with leading coefficient 1

Examples, conclusions and counter-examples

The following rings are principal ideal rings:

• Body
• ( the ring of integers )
• (the ring around the Gaussian numbers)
• The whole ring of the body, that is, the ring of Eisenstein numbers
• Polynomial in one variable over a field
• Formal power series rings in one indeterminate over a field
• Discrete valuation rings
• Euclidean rings (though this class includes all of the above examples, but not every principal ideal ring is Euclidean )
• Localizations of principal ideal rings are principal ideal rings again.

Principal ideal rings belong to the following general classes of rings:

• Factorial rings. In particular include:

• An element is exactly then prime if it is irreducible.
• Each non-zero element of the quotient of the body can be uniquely in the form of

• The lemma of Gauss: Every irreducible element is either an irreducible element of ( perceived as constant polynomial ) or into irreducible polynomial whose coefficients are relatively prime.

• Principal ideal rings are always Dedekind rings ( see also below)

No principal ideal rings are:

• The polynomial ring over the integers is not a principal ideal ring, because the ideal of 2 and obtained can not be generated by a single polynomial. This ring is but after the mentioned lemma of Gauss factorization, since it is a polynomial ring over a factorial ring.
• Is not a principal ideal ring, since the ideal ( x, y) is not a principal ideal.

Divisibility

• The largest (up to Associated awareness unique) common divisor of elements is the (unique up to Associated awareness ) generators of the ideal. In particular, the lemma of Bézout applies: There exist with

• The least common multiple of the generators of the ideal.
• Chinese Remainder Theorem: Are pairwise relatively prime, so is the canonical homomorphism

• A worsening of the Chinese remainder theorem is the approximation theorem: be Given pairwise distinct and numbers. Then there exists an approximating regarding in -order and is otherwise regular, ie

• For are equivalent:   is irreducible
• Is a prime element
• Is a prime ideal
• Is a maximal ideal

Principal ideal rings and Dedekind rings

Many naturally occurring rings in algebraic number theory and algebraic geometry are not principal ideal rings, but belong to a more general class of rings on the Dedekind rings. They are the localized version of the principal ideal rings, ideals are not globally but only locally generated by an element:

The following properties apply to principal ideal rings, but also more generally for Dedekind rings:

• They are either body or one-dimensional, ie every prime ideal is maximal equal.
• They are completely finished in its quotient field.
• They are regular.
• Your local rings are either body or discrete valuation rings.
• The above approximation theorem

Is a Dedekind ring is factorial or semi local, he is a principal ideal ring.

Modules over principal ideal rings

General

• Submodules of free modules are free.
• If a finitely generated module with Torsionsuntermodul, so there is a free submodule, so that. Torsion free, finitely generated modules are free.
• Projective modules are free.
• A module is injective if and only if it is divisible. Quotient injective modules are injective, each module has an injective resolution of length 1 is an explicit injective resolution of

Finitely generated modules: elementary partial replacement

The elementary part replacement describes the structure of a decomposition of a finitely generated module into indecomposable modules. ( A module is called irreducible if there are no modules with. )

It is above a system of representatives of irreducible elements (except for Associated awareness ). There is for every finitely generated module uniquely determined non-negative integers and, almost all of which are zero, so that

The numbers are determined by clear and the individual factors or are indecomposable. The ideals for which holds, hot elementary divisors of.

Finitely generated modules: invariant factors

There is finally generated for each module is a finite series of elements of which are not necessarily different from zero, so that

The ideals are determined by unique and called the invariant factors. The elements are thus uniquely determined up to Associated awareness.

It is added to this statement about modules, two competing perspectives:

• For a module, you can choose producers and look at the core of the associated homomorphism.
• At a sub-module, one can choose producers and the matrix with entries in look that describes the homomorphism with image.

Conversely, the image of a matrix with entries in a sub-module, and the quotient module ( the cokernel of the homomorphism given by ) is a finitely generated module.

For submodules of free modules is the statement:

• Is a free module and a ( also free ) submodule of the rank, so there are elements that are part of a base are, as well as elements, so that is a basis of. The spanned by the invariant part can be described as the archetype of the Torsionsuntermoduls of. The ideals are the invariants (as above ) of the module, possibly supplemented by.

For matrices (Smith normal form ):

• Is a matrix with entries in the rank of entries, so there are invertible matrices, so that has the following form:

Torsionsmoduln

There is a (not necessarily finitely generated ) torsion about, ie for each there exists a with. Again, it is a system of representatives of irreducible elements. Then: is the direct sum of - primary submodules, ie

With

As a corollary it follows that is semisimple if and only if for all.

Examples of use:

• If and so is the statement: Every rational number has a unique representation

• If ( a body), and so corresponds to the rational functions whose denominator is a power of. Thus, the set provides the first step of partial fractions, ie the unique representation of a rational function as

• Is and a finite-dimensional vector space together with an endomorphism ( with the module structure), the above decomposition is the splitting in the main rooms. The corollary implies in this case that is exactly then semisimple if the minimal polynomial of no multiple factors contains.

Related terms

• Conversely, it is an integral domain to represent the condition that all finitely generated ideals are principal ideals: These are the so-called Bézoutringe. Principal ideal rings are exactly the Noetherian Bézoutringe.
• Sometimes not divisors of zero rings are allowed in the definition of " principal ideal ring ", so it will only require that every ideal is a principal ideal and. In the English language can be made ​​between principal ideal ring and principal ideal domain ( domain = integral domain ), in German which is unusual.