Quasi-Euclidean subrings of □x

摘要

Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of □[x] which are not k-stage Euclidean for any norm and positive integer k. These subrings can be either PID or non-UFD, depending on the choice of parameters in our construction. In both cases, there are 2~ω such domains up to ring isomorphism.