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.



"Archimedeanising" an ordered field
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](http://math.haifa.ac.il/kobi/Wroclaw_proc.pdf), 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)


