[FOM] UFD Example

[Dave Marker]

Let F be a field and let M be an elementary submodel of F[[X,Y]]

Since F is an elementary submondel, M is still a UFD.

The ideal (X,Y) is still nonprinciple in M.

M is not a "polynomial ring" i.e. if R is a subring, then
M is not R[z] where z is transcendental over R.
Suppose not.  In F[[X,Y]] an element is a unit if and only
if the constant term is nonzero.  Thus either z or 1+z is
a unit in M and hence in R[z]. Both contradict that z is
transcendental over R.

Dave Marker