Search results
Results from the WOW.Com Content Network
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements p i of R: x = p 1 p 2 ⋅⋅⋅ p n with n ≥ 1. and this representation is unique in the following sense: If q 1, ..., q m are irreducible elements ...
Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
Download as PDF; Printable version ... In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique ... All free associative ...
There is a version of unique factorization for ordinals, though it requires some additional conditions to ensure uniqueness. Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic ...
In particular if k is a field, the ring of integers, or a principal ideal domain, then the polynomial ring [, …,] is regular. In the case of a field, this is Hilbert's syzygy theorem. Any localization of a regular ring is regular as well. A regular ring is reduced [b] but need not be an integral domain. For example, the product of two regular ...
Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals. [10] The units of a ring of integers O K is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units. [11]
This difficulty was resolved by Dedekind, who proved that the rings of algebraic integers have unique factorization of ideals: in these rings, every ideal is a product of prime ideals, and this factorization is unique up the order of the factors. The integral domains that have this unique factorization property are now called Dedekind domains ...