From the introduction: Let \(R\) be a Noetherian domain containing the rational number field \(\mathbb{Q}\) and let \(A=R[X]/I\), where \(R[X]\) is a polynomial ring in one variable over \(R\) and \(I\) is a nonzero prime ideal of \(R[X]\) with \(I\cap R=(0)\). In the earlier paper: Commun. Algebra 26, No. 7, 2033-2040 (1998; Zbl 0921.13002), N. Onoda, T. Sugatani and K. Yoshida considered the kernel \(C\) of the universal derivation \(d_{A/R}: A\to\Omega_R(A)\) and proved that \(C\) is Noetherian if and only if \(c(I)=R\), where \(c(I)\) denotes the content of \(I\); that is, \(c(I)\) is the ideal of \(R\) generated by the coefficients of the polynomiaIs in \(I\). The purpose of this article is to generalize this result to the case where \(C\) is the kernel of an \(R\)-derivation \(D:A\to M\) for some \(A\)-module \(M\).