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Right cones are semigroups with a left cancellation law such that for any two elements a, b there exists an element c with b s ac or a s 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 Holder's theorem. Right cones of rank 1 are classified. We then consider the problem of constructing for a given right cone H a right chain ring R with the same right ideal and ideal structure as H. ᮊ

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