Almost-Schreier Domains

Abstract

As an extension of the class of (pre)-Schreier domains introduced by P. M. Cohn and M. Zafrullah, we introduce and study a class of integral domains D characterized by the property that whenever a, b ₁, b ₂ ∈ D − {0} and a | b ₁ b ₂, there exist an integer k ≥ 1 and a ₁, a ₂ ∈ D − {0} such that a ^k  = a ₁ a ₂ and , i = 1, 2. We call them almost-Schreier domains. We show that an almost-Schreier domain has torsion t-class group, that a local (Noetherian) one-dimensional domain is almost-Schreier and that the polynomial ring with coefficients in an integrally closed almost-Schreier domain is almost-Schreier.

Key Words:

• Almost GCD domain
• Schreier domain

2000 Mathematics Subject Classification:

• Primary 13A15
• Secondary 13F05

ACKNOWLEDGMENTS

The first author gratefully acknowledges the warm hospitality of the Abdus Salam School of Mathematical Sciences GCU Lahore during his visits in 2006–2009. The second author was partially supported by an HEC (Higher Education Commission, Pakistan) grant.

Notes

Communicated by I. Swanson.