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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 $I = f(x)K[x] \cap R[x]$ 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 $I = f(x)K[x] \cap R[x]$ we have: – $I^{-1}\cap K[x]=(I:_{K(x)}I)$. – If there exists $p/q \in I^{-1}-K[x]$, then $(q,f)\neq 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I' = q(x)K[x] \cap R[x]$ 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.
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