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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 ...
Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains. 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 ...
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.
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another ...
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 ...
More technically, a greatest common divisor of a and b is a generator of the ideal generated by a and b (this characterization is valid for principal ideal domains, but not, in general, for unique factorization domains). The greatest common divisor of two Gaussian integers is not unique, but is defined up to the multiplication by a unit.
P.M. Cohn, "Noncommutative unique factorization domains", Transactions of the American Mathematical Society 109:2:313-331 (1963). full text R. Sivaramakrishnan, Certain number-theoretic episodes in algebra , CRC Press, 2006, ISBN 0-8247-5895-1
Unlike principal ideal domains (where every ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of entire functions is a non-atomic Bézout domain, and there are many other examples. An integral domain is a Prüfer GCD domain if and only if it is a Bézout domain. [3]
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