- 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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