Almost-Schreier Domains

As an extension of the class of (pre)-Schreier domains introduced by P.M. Cohn and M. Zafrullah we introduce and study 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 almost-Schreier monoids.