A nonzero and noninvertible element a of a ring R which generates a prime ideal. It can also be characterized by the condition that whenever a divides a product in R, a divides one of the factors. The prime elements of Z are the prime numbers P. In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring Z[isqrt(5)], 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 (1 - isqrt(5))(1 + isqrt(5)) = 6, but it does not divide any of the factors.

irreducible element | prime number | unique factorization | unique factorization domain