  In this paper we study the set of prime ideals in vector lattices and how the
properties of the prime ideals structure the vector lattice in question. The
different properties that will be considered are firstly, that all or none of
the prime ideals are order dense, secondly, that there are only finitely many
prime ideals, thirdly, that every prime ideal is principal, and lastly, that
every ascending chain of prime ideals is stationary (a property that we refer
to as prime Noetherian). We also completely characterize the prime ideals in
vector lattices of piecewise polynomials, which turns out to be an interesting
class of vector lattices for studying principal prime ideals and ascending
chains of prime ideals.

Commutative Algebra;Functional Analysis