Abstract
We show how to construct discretely ordered principal ideal sub rings of Q[x] with various types of prime behaviour. Given any set V consisting of finite strictly increasing sequences (d(1), d(2), . . . , d(l)) of positive integers such that, for each prime integer p, the set {pZ, d(1)+pZ, . . . , d(l)+pZ} does not contain all the cosets modulo p, we can stipulate to have, for each (d(1), . . . , d(l)) is an element of D, a cofinal set of progressions (f, f + d(1), . . . , f + d(l)) of prime elements in our principal ideal domain R-tau.
Moreover, we can simultaneously guarantee that each positive prime g is an element of R-tau \ N is either in a prescribed progression as above or there is no other prime h in R tau such that g - h is an element of Z. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring (Z) over cap of profinite integers.