Ideal theory of right cones and associated rings

Right cones are semigroups with a left cancellation law such that for any t wo elements a, b there exists an element c with b = ac or a = be. Valuation rings, cones of ordered or left ordered groups, semigroups of ordinal numb ers, and Hjelmslev rings are examples. The ideal theory of these semigroups is described in terms of prime and completely prime ideals, and a classifi cation of prime segments is given that can be used to solve a problem raise d by Skornyakov. The Archimedean case can be dealt with in a satisfactory w ay 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 ri ght chain ring R with the same right ideal and ideal structure as H. (C) 19 98 Academic Press.