abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, a Schreier domain, named after

Otto Schreier Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. Life His parents were the arc ...

, is an

integrally closed domain In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...

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 In mathematics, a GCD domain is an integral domain ''R'' with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalentl ...

is an example of a Schreier domain. The term "Schreier domain" was introduced by

P. M. Cohn Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commuta ...

in 1960s. The term "pre-Schreier domain" is due to Muhammad Zafrullah. In general, an

irreducible element In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements. Relationship with prime elements Irreducible elements should not be confus ...

is primal if and only if it is a

prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...

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

unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...

; 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. Ring theory {{Abstract-algebra-stub