Building Discretely Ordered Bezout Domains and GCD Domains Original Research Article

Using constructions due to A. Macintyre and D. Marker, we build GCD domains and Bezout domains with the open induction property. In fact we show that an open induction domain can be a principal ideal domain different from image. The rings we construct are all countable or of cardinality aleph, Hebrew1; we show that the order type of the infinite primes is arbitrary for GCD domains, subject to this cardinality restriction. This result also holds for countable Bezout domains. Our structures all have the additional property that any nonzero element is divisible by only finitely many n set membership, variant image.