If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x \in K$ such that $|x| \le 1/n$ for every non-zero $n \in N$). Then $I$ is a maximal ideal and the order on $K$ induces an order on $A = B/I$ making it into an archimedean ordered field.

Has this construction been studied? More specifically, when does the natural projection of $B$ onto $A$ split (in the category of rings or, better still, ordered rings)? I believe it always has an order-preserving splitting if $K$ is real closed.

  • $\begingroup$ Can you just choose a transcendence basis for $A$ over $\mathbb Q$, for each one pick an arbitrary point in their inverse image, and then for every other number pick the unique point in its inverse image which satisfies its minimal polynomial over $\mathbb Q$ adjoin the transcendence basis? $\endgroup$ – Will Sawin Feb 3 '12 at 18:51
  • $\begingroup$ That doesn't work in general. Let $K$ be the splitting field for the polynomial $X^2 - 2 - \epsilon$ over the field $\mathbb{Q}(\epsilon)$ of rational functionals with rational coefficients of an an infinitesimal indeterminate $\epsilon$. Then $A$ is isomorphic to $\mathbb{Q}[\sqrt{2}]$, but 2 is not a square in $K$. $\endgroup$ – Rob Arthan Feb 4 '12 at 9:33

Yes, the map is sometimes called the standard part map, and it turns out that one map define it in general for a group in an o-minimal structure. See here, for example. Regarding the splitting, if it should only respect the additive group structure, I think it exists since everything is a vector space over $\mathbb{Q}$ (as explained in the comment)

  • $\begingroup$ But I guess the splitting requires the Axiom of Choice. $\endgroup$ – Gerald Edgar Feb 3 '12 at 22:30
  • $\begingroup$ @Moshe: Thanks for a useful reference. I think you will find the existence of a standard part map is a hypothesis rather than a theorem in the work you refer to e.g., see the paper by Marikova. I should have said that the splittings I am interested in are to be ring homomorphism, or, even better, ordered ring homomorphisms. As you say the projection always splits when you just view $K$ and $A$ as vector spaces over $\mathbb{Q}$ (given AC to construct a basis, in general, as Gerald points out). $\endgroup$ – Rob Arthan Feb 4 '12 at 9:46
  • $\begingroup$ @Moshe: Apologies: I misunderstood your reply. I think now that you were saying that the projection is called the "standard part map" and I have no quibbles with that. $\endgroup$ – Rob Arthan Feb 4 '12 at 13:34
  • $\begingroup$ Yes, that is what I meant. A nice point is that it is possible to define it for arbitrary groups in an o-minimal theory, and it turns out that (at least if the group is compact in a suitable sense) the quotient is a Lie group (this is Pillay's conjecture, which is now a theorem, and is explained in the above notes). As for the ring theoretic splitting, I don't think you can have it, since it would have to be continuous, but the topology on K can be pretty bad. $\endgroup$ – Moshe Feb 7 '12 at 14:32

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