what kind of integral domain do the non-infinite surreals form? https://en.wikipedia.org/wiki/Integral_domain
mentions the following chain of inclusions:
Principal Ideal domains $\subset$ Unique Factorization domains $\subset$ GCD domains $\subset$ Integrally Closed domains
Do the non-infinite surreals (i.e. those that are either infinitesimal or finite) form an I.C. domain? Do they form a GCD domain? A U.F. domain? A P.I. domain?
 A: It's a local ring whose unique maximal ideal is the class of infinitesimal elements.
In fact, I believe for any two ideals, you have either $I \subseteq J$ or $J \subseteq I$. (thus, you have a GCD domain)
All of the non-infinitesimal elements are units, so the only nontrivial aspect of 'factorization' is how infinitesimal an element is. e.g. $\epsilon^2$ is more infinitesimal than $\epsilon$, as can be seen by observing $\frac{\epsilon^2}{\epsilon}$ is infinitesimal, or equivalently, that $\frac{\epsilon}{\epsilon^2}$ is infinite.
The maximal ideal is certainly not principal; every ideal $(\epsilon)$ fails to contain the element $|\epsilon|^{1/2}$.
(these are true things for any real closed field with infinite elements, which I believe the surreals are, barring size issues)
I imagine the "degree" of infinitesimallitude is measured again by positive surreal numbers; e.g. given any two sets $L$ and $R$ of infinitesimals where everything in $L$ is less infinitesimal than everything in $R$, there should be infinitesimals strictly between $L$ and $R$. But I'm not experienced with this, so take my claim with a grain of salt.
