AbstractAn additive subgroupPof a skew fieldFis called aprimeofFifPdoes not contain the identity, but if the productxyof two elementsxandyinFis contained inP, thenxoryis inP. A prime segment ofFis given by two neighbouring primesP1⊃P2; such a segment is invariant, simple, or exceptional depending on whetherA(P1) = {a∈P1|P1aP1⊂P1} equalsP1,P2or lies properly betweenP1andP2. The setT(F) of all primes ofFtogether with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fieldsFof fractions of group rings of certain right ordered groups. In particular, groupsGof affine transformations on ordered vector spacesVare considered, and the relationship between properties of Dedekind cuts ofV, certain right orders onG, and chains of prime segments ofFis investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.