Ideal-theoretic characterizations of valuation  and Pr\"ufer monoids

Franz Halter-Koch

Institut f\"ur Mathematik, Karl-Franzens-Universit\"at Graz
Heinrichstrasse 36, 8010 Graz, Austria


It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system
(Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and
apply it to give several new characterizations of Pr\"ufer (multiplication) monoids and integral domains.

AMSclassification. Primary: 20M14; Secondary: 13A15.

Keywords.  Valuation monoids, Pr\"{u}fer domains.