Uppers to zero in R[x] and almost principal ideals

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in Kx]. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f(x)Kx] ∩ Rx] are almost principal in the following two cases:

• J, the ideal generated by the leading coefficients of I, satisfies J ^?1 = R.
• I ^?1 as the Rx]-submodule of K(x) is of finite type.

Furthermore we prove that for I = f(x)Kx] ∩ Rx] we have:

• I ^?1 ∩ Kx] = (I: K(x) I).
• If there exists p/q ∈ I ^?1 ? Kx], then (q, f) ≠ 1 in Kx]. If in addition q is irreducible and I is almost principal, then I′ = q(x)Kx] ∩ Rx] is an almost principal upper to zero.

Finally we show that a Schreier domain R is a greatest common divisor domain if and only if every upper to zero in Rx] contains a primitive polynomial.