Content deleted Content added
better |
No edit summary |
||
Line 47:
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 ___domain | Noetherian ring]] 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.
| |||