Schreier domain

Short description: Mathematical structure where elements are primal

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 = x₁ x₂ so that x₁ divides y and x₂ 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, 1968.
• Zafrullah, Muhammad, On a property of pre-Schreier domains, 1987.