Prime ideal need not contain any prime element

Statement

It is possible to have an integral domain $R$ and a nonzero prime ideal $P$ of $R$ such that $P$ does not contain any prime element of $R$.

Related facts

For unique factorization domains instead of arbitrary integral domains (strengthening of hypothesis)

• Unique factorization implies every nonzero prime ideal contains a prime element
• Unique factorization and one-dimensional iff principal ideal
• Unique factorization and finite-dimensional implies every prime ideal is generated by a set of primes of size at most the dimension
• Unique factorization and Noetherian implies every prime ideal is generated by finitely many prime elements

For irreducible elements instead of prime elements (weakening of conclusion)

• ACCP implies every nonzero prime ideal contains an irreducible element
• Noetherian domain implies every prime ideal is generated by finitely many irreducible elements

Proof

For an example, we can take any Dedekind domain that is not a principal ideal domain, and pick a prime ideal in the Dedekind domain that is not principal. A concrete example is:

$R=\mathbb {Z} [{\sqrt {-5}}],P=(2,1+{\sqrt {-5}})$.