Ideal generated by two prime elements in a unique factorization domain may be proper and not prime
It is possible to have a unique factorization domain and primes such that is a proper ideal of that is not prime.
- Unique factorization is not prime-quotient-closed: If is a unique factorization domain, and is a prime ideal, the quotient need not be a unique factorization domain. Examples of this where the prime ideal is principal also give examples of ideals generated by two elements that are proper and not prime.
- Unique factorization implies every prime ideal is generated by prime elements
- Unique factorization implies every nonzero prime ideal contains a prime element
- Unique factorization and Noetherian implies every prime ideal is generated by finitely many prime elements
Example of a bivariate polynomial ring
Let be any field of characteristic not equal to . Let <mah>R = k[x,y]</math>. The polynomials and are irreducible in , and hence prime. However, the ideal is not prime in . To see this, note that , which is not an integral domain, because is not irreducible.
Example of a polynomial ring over the integers
Let be the ring of rational integers and . The elements are irreducible in . However, the ideal is not a prime ideal in . To see this, note that being prime is equivalent to the polynomial being irreducible in , which is not true since the polynomial reduces as modulo .