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.