Unique Factorization Domain

In Mathematics , a unique factorization domain (UFD) is, roughly speaking, a Commutative Ring in which every element can be uniquely written as a product of Prime Element s, analogous to the Fundamental Theorem Of Arithmetic for the Integer s. UFDs are sometimes called '''factorial rings''', following the terminology of Bourbaki .

Formally, a unique factorization domain is defined to be an Integral Domain ''R'' in which every non-zero non- Unit ''x'' of ''R'' can be written as a product of Irreducible Element s of ''R'':

x

and this representation is unique in the following sense: if ''q''₁,...,''q''_''m'' are irreducible elements of ''R'' such that

x

then ''m'' = ''n'' and there exists a to ''q''_φ(''i'') for ''i'' = 1, ..., ''n''.

The uniqueness part is sometimes hard to verify, which is why the following equivalent definition is useful: a unique factorization domain is an integral domain ''R'' in which every non-zero non-unit can be written as a product of Prime Element s of ''R''.

EXAMPLES

Most rings familiar from elementary mathematics are UFD's:

the Integer s. This is the fundamental theorem of arithmetic.

any Field ; this includes the fields of Rational Number s, Real Number s, and Complex Number s.

If ''R'' is a UFD, then so is ''R'' {Link without Title} , the Ring Of Polynomials with coefficients in ''R''. A special case of this, due to the above, is that the polynomial ring over any field is a UFD.

Here are some more examples of UFDs:

The Gaussian Integer s, Z {Link without Title} .

The Formal Power Series ring ''K'' ''X''₁,...,''X''_''n'' over a field ''K''.

The ring of functions in ''n'' complex variables Holomorphic at the origin is a UFD.

Despite these examples, very few integral domains are UFDs. Here is a counterexample:

The ring $\mathbb Z$ {Link without Title} of all complex numbers of the form $a+b\sqrt{-5}$ , where ''a'' and ''b'' are integers. Then 6 factors as both (2)(3) and as $\left(1+\sqrt{-5}
ight)\left(1-\sqrt{-5}
ight)$ . These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, $1+\sqrt{-5}$ , and $1-\sqrt{-5}$ are Associate . It is not hard to show that all four factors are irreducible as well, though this may not be obvious. See also Algebraic Integer .

Most Factor Ring s of a polynomial ring are not UFDs. Here is an example:

Let ''R'' be any commutative ring. Then ''R'' {Link without Title} /(''XY''−''ZW'') is not a UFD. It is clear that ''X'', ''Y'', ''Z'', and ''W'' are all irreducibles, so the element ''XY''−''ZW'' has two factorizations into irreducible elements.

PROPERTIES

Additional examples of UFDs can be constructed as follows:

All Principal Ideal Domain s are UFDs.

If ''R'' is a UFD, then so is the polynomial ring ''R'' By Induction , we can show that the polynomial rings Z[''X''₁, ..., ''X''_''n'' as well as ''K'' ..., ''X''_''n'' (''K'' a field) are UFD's. (Any polynomial ring with more than one variable is an example of a UFD that is not a principal ideal domain.)

Some concepts defined for integers can be generalized to UFDs:

In UFD's, every Irreducible Element is Prime . (In any integral domain, every prime element is irreducible, but the converse does not always hold.)

Any two (or finitely many) elements of a UFD have a Greatest Common Divisor and a Least Common Multiple . Here, a greatest common divisor of ''a'' and ''b'' is an element ''d'' which Divides both ''a'' and ''b'', and such that every other common divisor of ''a'' and ''b'' divides ''d''. All greatest common divisors of ''a'' and ''b'' are associated.

EQUIVALENT CONDITIONS FOR A RING TO BE A UFD

Under some circumstances, it is possible to give 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.

An integral domain is a UFD if and only if the ascending chain condition holds for principal ideals, and any two elements of ''A'' have a least common multiple.

A ring is a UFD if and only if its Class Group is zero.

	CATEGORIES ABOUT UNIQUE FACTORIZATION DOMAIN

	abstract algebra

	ring theory

	number theory

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Unique Factorization Domain