**Unique Factorization Domain**

In mathematics, a **unique factorization domain (UFD)** is a commutative ring in which every non-unit element, with special exceptions, can be uniquely written as a product of prime elements (or irreducible elements), analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called **factorial rings**, following the terminology of Bourbaki.

Note that unique factorization domains appear in the following chain of class inclusions:

**Commutative rings**⊃**integral domains**⊃**integrally closed domains**⊃**unique factorization domains**⊃**principal ideal domains**⊃**Euclidean domains**⊃**fields**

Read more about Unique Factorization Domain: Definition, Examples, Non-examples, Properties, Equivalent Conditions For A Ring To Be A UFD

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*K*- Factorization in

*K*

... is less than the degree of q, and a decomposition with these properties is

**unique**... Rings for which there exists

**unique**(in an appropriate sense)

**factorization**of nonzero elements into irreducible factors are called

**unique factorization domains**or factorial rings the given construction shows that all ... Thus the polynomial ring K is a principal ideal

**domain**, and for the same reason every Euclidean

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**domain**...

**Unique Factorization Domain**- Equivalent Conditions For A Ring To Be A UFD

... A Noetherian integral

**domain**is a UFD if and only if every height 1 prime ideal is principal ... Also, a Dedekind

**domain**is a UFD if and only if its ideal class group is trivial ... In this case it is in fact a principal ideal

**domain**...

... A

**unique factorization domain**is a GCD

**domain**... Among the GCD

**domains**, the

**unique factorization domains**are precisely those that are also atomic

**domains**(which means that at least one

**factorization**into irreducible elements exists for any nonzero nonunit) ... A Bézout

**domain**(i.e ...

... If R is an integral

**domain**, an element f of R which is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh ... is irreducible the converse is not true in general but holds in

**unique factorization domains**... The polynomial ring F over a field F (or any

**unique**-factorization

**domain**) is again a

**unique factorization domain**...

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