Ideal Theory of Right Cones and Associated Rings

Right cones are semigroups with a left cancellation law such that for any two elementsa, bthere exists an elementcwithb = acora = bc. Valuation rings, cones of ordered or left ordered groups, semigroups of ordinal numbers, and Hjelmslev rings are examples. The ideal theory of these semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given that can be used to solve a problem raised by Skornyakov. The Archimedean case can be dealt with in a satisfactory way with the help of Hölderʹs theorem. Right cones of rank 1 are classified. We then consider the problem of constructing for a given right coneHa right chain ringRwith the same right ideal and ideal structure asH.