Almost-Schreier Domains

Abstract

In this thesis, 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, b1 , b2 ∈ D − {0} and a|b1 b2 , there exist an integer k ≥ 1 and a1 , a2 ∈ D − {0} such that ak = a1 a2 and ai |bk , i = 1, 2. i 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. Then we convert a great part of the material about almost-Schreier domains in the language of monoids, that is, we introduce and study the cancellative monoids H characterized by the same property as above.