In algebra, a Schreier domain is an integrally closed

Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings. -Commutative rings:...

 integral 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

GCD domain

In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor . Equivalently, any two non-zero elements of R have a least common multiple ....

 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

Irreducible element

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...

 is primal if and only if it is a prime element

Prime element

In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...

. Consequently, in a Schreier domain, every irreducible is prime. In particular, an atomic

Atomic domain

In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain, every non-zero non-unit of which can be written as a product of irreducible elements...

 Schreier domain is a unique factorization domain

Unique factorization domain

In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...

; this generalizes the fact that an atomic GCD domain is a UFD.