An integral domain $D$ is a pre-Schreier domain if every non-zero element of $D$ is primal. If in addition $D$ is integrally closed, then $D$ is called a Schreier domain.

Remarks.

1. Every irreducible element of a pre-Schreier domain is prime.
2. A gcd domain is a Schreier domain (a proof of this can be found here).