On the kernels of universal derivations and extensions \(R[\alpha,1/\alpha]\) of \(R\) (Q2710510)

Let \(R\) be a Noetherian domain, \(x\) an algebraic element over the fraction field \(K\) of \(R\), \(f\) its minimal polynomial over \(K\): \(f:= \text{Irr}(x,K)=X^d+a_1X^{d-1}+\ldots +a_d\), \(a_i\in K\), and \(h:R[X]\rightarrow R[x]\) the map \(X\rightarrow x\). Let \(I=\bigcap_{i=1}^d (R:_R a_i)\) and \(J=I(1,a_1,\ldots ,a_d)\) and suppose that \(\operatorname {Ker} h=If R[X]\) [if \(R\) is Krull then this condition is fulfilled by any algebraic element \(x\) over \(R\) by a previous authors' result; see \textit{S. Oda, J. Sato} and \textit{K. Yoshida}, Osaka J. Math. 30, No. 1, 119-135 (1993; Zbl 0782.13013)]. If \(B:= R[x,1/x]\) is unramified over \(R\) then \(B\) is flat over \(R\). If \(J=R\) and \(\mathbb{Q}\subset R\) then the kernel of the universal derivation \(B\rightarrow\Omega_R(B)\) is a finitely generated \(R\)-algebra.