Prime Element

A nonzero and noninvertible element of a ring which generates a prime ideal. It can also be characterized by the condition that whenever divides a product in , divides one of the factors. The prime elements of are the prime numbers .

In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring , where i is the imaginary unit, is not a unique factorization domain, and there the element 2 is irreducible, but not prime, since 2 divides the product , but it does not divide any of the factors.