Quasi-Euclidean subrings of Q[x]

Abstract

Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of Q[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^\omega such domains up to ring isomorphism.

Keywords

• k-stage division chain
• PID
• Quasi-Euclidean domain
• Primary 13F07
• Secondary 13F10
• 13F20
• 03H15