Unique factorization domain

A factorial ring, and ZPE -ring ( Short for: " The decomposition into primes is unique. " ), Gaussian ring or EPC - ring is an algebraic structure, namely an integral domain in which every element has a substantially unique decomposition into irreducible factors has. factorial rings are not to be confused with factor rings.

Definition

An integral domain is called factorial if it has the following property:

• Each element has a unique up to Assoziierheit and order decomposition into irreducible factors.

For an integral domain, the property of being factorial, is equivalent to the property of being a ZPE ring:

• Each element has a decomposition into a product of prime elements. ( Representations as a product of prime elements are clearly always essentially. )

Decomposition into irreducible factors

Has a decomposition into irreducible factors, when a is a representation

Has a unit and irreducible elements. It is the empty product of irreducible elements, ie, approved, which is equivalent to the identity element of the ring. This decomposition is essentially unique if for each additional such representation

Where: and (after eventual renumbering ).

Means and are associated.

Are not only irreducible but even prime elements, already follows from the uniqueness of the representation (except for Associated awareness ).

Properties

• Irreducible elements in factorial rings are prime. ( This also follows the equivalence of the above descriptions. )
• By the lemma of Gauss polynomial rings are factorial factorial again.

Examples

• Every Euclidean ring is a principal ideal ring, and each principal ideal ring is a factorial ring. Examples are the Euclidean rings ( whole numbers ) and the polynomial ring in one variable over a field.
• Conversely, however, not everyone factorial ring automatically principal ideal ring. The wholeness rings of algebraic number fields, however, the two concepts coincide.
• Although bodies have not yet primes irreducible elements, but are also factorial rings, since each non-zero element of the body is a unit.
• Polynomial rings and rings of formal power series over a field
• Polynomial rings over a factorial ring is again factorial ( Gauss, see Content ( polynomial ) )
• Regular local rings

Counter-examples

An example of a ring in which there is a separation into irreducible elements is not unique, the ring (see the adjunction ): In the two product images

Are the irreducible factors in each case, but among the four numbers and no two are associated. The units in this ring are and.

An example of a ring in which a decomposition into irreducible elements do not always exist, but it is unique whenever it exists, is the ring of holomorphic functions on a region in the complex plane ( with pointwise addition and multiplication ): This ring is zero divisors (which follows from the identity theorem for holomorphic functions). The units of the holomorphic functions without zeros are exactly ( eg the complex exponential ). The irreducible elements are up to units exactly the functions of the form () for a point. It follows that a holomorphic function if and only a product of irreducible elements if it has only finitely many zeros. But since in each area also are holomorphic functions with infinitely many zeros, this ring is not a factorial ring. If a holomorphic function, however, has such a representation, so this is clearly mainly because the irreducible elements are all prime.