Unique factorization ___domain: Difference between revisions

Revision as of 06:14, 14 April 2008

Revision as of 06:16, 14 April 2008

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Some concepts defined for integers can be generalized to UFDs:

* In UFDs, every [[irreducible element]] is [[prime element|prime]]. (In any integral ___domain, every prime element is irreducible, but the converse does not always hold.) Note that this has a partial converse: any [[Noetherian ___domainring | Noetherian ring___domain]] is a UFD iff every irreducible element is prime (this is one proof of the implication PID <math>\Rightarrow</math> UFD).

* 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 [[divisor|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.