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

作者：Borna Keivan^*; Mohajer Naser Abolfazl

来源：Czechoslovak Mathematical Journal, 2013, 63(2): 565-572.

DOI：10.1007/s10587-013-0038-9

• almost principal ideal
• divisorial ideal
• greatest common divisor domain
• Schreier domain
• uppers to zero

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in K[x]. 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)K[x] boolean AND R[x] 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 R[x]-submodule of K(x) is of finite type.
Furthermore we prove that for I = f(x)K[x] boolean AND R[x] we have:
I (-1) boolean AND K[x] = (I: K(x) I).
If there exists p/q epsilon I-1 - K[x], then (q, f) not equal 1 in K[x]. If in addition q is irreducible and I is almost principal, then I' = q(x)K[x] boolean AND R[x] 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 R[x] contains a primitive polynomial.

• 出版日期2013-6