Prime element
From AoPSWiki
In ring theory an element 
of an integral domain 
is said to be prime if:
- is not a unit.
- If
for any 
then 
or 
.
Equivalently, we can say that is prime iff 
is a prime ideal in .
Any prime element 
is clearly irreducible in . (Indeed if 
, then we would have , so would have to divide one of 
and 
, WLOG . Then 
for some 
, so 
, so 
, and hence is a unit.) The converse of this holds in any unique factorization domain, but it does not hold in a general integral domain.
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