Polynomial Behavior of Prime Ideals in Polynomial Rings and the Projective Line Over Z

This chapter analyzes prime spectra of rings related to polynomial rings using the concept of a coefficient subset of a partially ordered set. Loosely speaking, a coefficient subset behaves like the set of prime ideals extended from a coefficient ring in the ring of polynomials over the coefficient ring. The chapter defines the projective line over Z and gives an analogous description of the projective line over Z. It also introduces a property that some subset of primes should behave like the extensions of primes of the coefficient ring. Such subsets are called coefficient sets.