An example of an ordered UFD except the ring of integers?

Question

Are there examples of unique factorization domains which are ordered rings

https://en.m.wikipedia.org/wiki/Ordered_ring

except the ring of integers?

Answer 1

Try $R=\Bbb Z[X]$ with $0<X\ll 1$ (i.e., $f>0\iff\exists \epsilon>0\colon \forall x\in(0,\epsilon)\colon f(x)>0$)

Answer 2

Any ordered field, like $\Bbb Q$ or $\Bbb R$, is also an ordered UFD.

For a more "interesting" family of examples, the polynomial ring over any ordered UFD in one variable is an ordered UFD, where an element is considered positive iff its leading coefficient is positive (and $f<g$ iff $0< g-f$). Or, as Hagen von Eizen suggests in his answer, an element is considered positive iff its lowest-degree non-zero coefficient is positive.