A prime element of a domain is an element which is neither zero nor a unit divisible only by units and associates and which also satisfies the following condition: if , where and are also in the same domain, then either or , maybe both; but if neither of those holds true, then may be irreducible but it is not prime. In fact, if the domain is not a unique factorization domain, it does not have prime elements though it may have irreducible elements.
For example, in (see: Gaussian integers), we see that and that . Though , this does not detract from the fact that is a prime element of .
Although 2 is irreducible in , it is not prime. For example, , but and either. Note that 6 has two factorizations into irreducibles: .
- ↑ Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 87