Euclid's lemma

Etymology
Named after ancient Greek mathematician (fl. 300 BCE). A version of the proposition appears in Book VII of his.

Noun

 * 1)  The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both; slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;  the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.

Usage notes
The proposition as generalised to principal ideal domains is occasionally called Gauss's lemma; some writers, however, consider this usage erroneous as another result is known by that term.