## Convexity, valuations and Prüfer extensions in real algebra

### Summary

Summary: We analyse the interplay between real valuations, Prüfer extensions and convexity with respect to various preorderings on a given commutative ring. We study all this first in preordered rings in general, then in $f$-rings. Most often Prüfer extensions and real valuations abound whenever a preordering is present. The next logical step, to focus on the more narrow class of real closed rings, is not yet taken, except in some examples.

### Mathematics Subject Classification

13J30, 13A18, 13F05

### Keywords/Phrases

valuation, Prüfer extension, real holomorphy ring, convex subring