# Schreier domain

In abstract algebra, a **Schreier domain**, named after Otto Schreier, is an integrally closed domain where every nonzero element is **primal**; *i.e.*, whenever *x* divides *yz*, *x* can be written as *x* = *x*_{1} *x*_{2} so that *x*_{1} divides *y* and *x*_{2} divides *z*. An integral domain is said to be **pre-Schreier** if every nonzero element is primal. A GCD domain is an example of a Schreier domain. The term "Schreier domain" was introduced by P. M. Cohn in 1960s. The term "pre-Schreier domain" is due to Muhammad Zafrullah.

In general, an irreducible element is primal if and only if it is a prime element. Consequently, in a Schreier domain, every irreducible is prime. In particular, an atomic Schreier domain is a unique factorization domain; this generalizes the fact that an atomic GCD domain is a UFD.

## References

- Cohn, P.M., Bezout rings and their subrings, 1967.
- Zafrullah, Muhammad, On a property of pre-Schreier domains, 1987.