Factorial ring

A factorial ring , also ZPE-ring (abbreviation for:. " Z Erlegung in P rimelemente is e indeutig"), Gaussian ring or EPZ-ring is an algebraic structure , namely an integrity ring , in which each element has a substantially unique decomposition in irreducible factors. Factorial rings are not to be confused with factor rings . ${\displaystyle a\neq 0}$

definition

An integrity ring is called factorial if it has the following property: ${\displaystyle A}$

• Each element has a clear decomposition into irreducible factors, except for association and sequence . ${\displaystyle a\neq 0}$

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

• Every element that is not a unit has a decomposition into a product of prime elements . (Representations as the product of prime elements are always essentially unique in integrity rings.) ${\displaystyle a\neq 0}$

Decomposition into irreducible factors

${\displaystyle a\in R}$ has a decomposition into irreducible factors if a is a representation

${\displaystyle a=\varepsilon \,q_{1}\,q_{2}\dots q_{r}}$

with unity and irreducible elements . The empty product of irreducible elements, i.e., which is to be equated with the one element of the ring , is permitted. This decomposition is essentially unambiguous if every further such representation ${\displaystyle \varepsilon }$ ${\displaystyle q_{i}}$ ${\displaystyle r=0}$

${\displaystyle a=\varepsilon '\,q_{1}'\,q_{2}'\dots q_{r'}'}$

applies: and (after possibly renumbering). ${\displaystyle r=r'}$ ${\displaystyle q_{i}\sim q_{i}'}$

${\displaystyle q_{i}\sim q_{i}'}$means: and are associated . ${\displaystyle q_{i}}$ ${\displaystyle q_{i}'}$

If they are not only irreducible, but also prime elements, the uniqueness of the representation follows from this (apart from associated). ${\displaystyle q_{1},q_{2},\dotsc ,q_{r}}$

properties

• Irreducible elements in factorial rings are prime. This also follows the equivalence of the descriptions given above.
• In factorial rings, every ascending chain of principal ideals becomes stationary. Conversely, if every ascending chain of main ideals in an integrity ring becomes stationary and every irreducible element there is a prime element, then it is a factorial ring.
• Factorial rings are GCD rings . After choosing a system of representatives for the prime elements, a greatest common divisor of finitely many elements can be calculated as the product of the common prime factors of these elements, taking into account the multiplicity.
• Factorial rings are normal , ie completely closed in the quotient field .
• According to Gauss' lemma , polynomial rings of factorial rings are factorial again.
• Locations of factorial rings are factorial (except when the null element is inverted).

Examples

• Every Euclidean ring is a main ideal ring , and every main ideal ring is a factorial ring. Examples are the Euclidean rings ( whole numbers ) and the polynomial ring in a variable over a body . ${\displaystyle \mathbb {Z} }$ ${\displaystyle K[X]}$ ${\displaystyle K}$
• Conversely, however, not every factorial ring is automatically the main ideal ring: The rings and are factorial, but not main ideal rings. However, the two terms coincide with the totality rings of algebraic number fields. ${\displaystyle K[X,Y]}$ ${\displaystyle \mathbb {Z} [X]}$
• Bodies have neither irreducible elements nor prime elements, but are also factorial rings, since every non-zero element of a body is a unit.
• The vast majority do not consider the null ring to be a factorial ring. Although the condition of the existence of a prime factorization is empty, the zero ring is not regarded as an integrity ring.
• Polynomial rings and rings of formal power series over a field are factorial.
• Regular local rings (e.g. discrete evaluation rings ) are factorial. This is exactly the statement of the Auslands-Boxwood theorem.

Counterexamples

An example of a ring in which there is a decomposition into irreducible elements that is not unique is the ring (see adjunction ): In the two product representations ${\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}$

${\displaystyle 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\cdot \left(1-{\sqrt {-5}}\right)}$

the factors are each irreducible, but among the four numbers and no two are associated. The units in this ring are and . ${\displaystyle 2,3,1+{\sqrt {-5}}}$ ${\displaystyle 1-{\sqrt {-5}}}$ ${\displaystyle +1}$ ${\displaystyle -1}$

An example of a ring in which a decomposition into irreducible elements does not always exist, but this is unique whenever it exists, is the ring of holomorphic functions on a domain in the complex plane (with pointwise addition and multiplication): This ring is zero-divisor-free (this follows from the identity theorem for holomorphic functions ). The units are exactly the holomorphic functions without zeros (e.g. the complex exponential function ). The irreducible elements are the functions of the form ( ) for a point with the exception of units . From this it follows that a holomorphic function is a product of irreducible elements if and only if it has a finite number of zeros. But since there are also holomorphic functions with an infinite number of zeros in every area, this ring is not a factorial ring. However, if a holomorphic function has such a representation, it is essentially unique because the irreducible elements are all prime. ${\displaystyle U}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z\mapsto z-a}$ ${\displaystyle a\in U}$

Individual evidence

1. ^ Serge Lang: Algebra . 3. Edition. Springer, 2008, ISBN 978-0-387-95385-4 , pp. 111 .
2. ^ Christian Karpfinger, Kurt Meyberg: Algebra. Group rings body. 3. Edition. Springer-Verlag, Berlin / Heidelberg 2013, sentence 17.1