If k is a field, k[x] is a
principal ideal domain and the ideal structure of k[x] is well understood. For
example, a nonzero ideal is prime if and only if its generator is irreducible. If R is an
integral domain with quotient field k, it is natural to ask if the set of ideals I of
R[x] such that Ik[x] is proper can be equally well described. Since such
ideals can contain no nonzero elements of R, one hopes that the structure
will be dominated by the structure of k[x]. While such ideals need not be
principal, we define a notion of almost principal which does hold for a large
class of rings R. We study this class and give examples where ideals are not
almost principal. The almost principal property is related to the following
questions:
When is (ax − b)k[x] ∩ R[x] generated by linear elements? (Ratliff)
When is (f(x))k[x] ∩ R[x] divisorial? (Houston) and
When is an ideal I, which is its own extension-contraction from R[x] to
R[[x]] and back, equal to cl(I) in the x-adic topology? (Arnold)
