# Prime ideal need not contain any prime element

From Commalg

## Contents

## Statement

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

## 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:

.