The Laurent extension of a noetherian integral domain. (Q2731716)

Let \(R[\alpha,~\alpha^{-1}]\) be an extension of a noetherian integral domain \(R\) where \(\alpha\) NEWLINEis an element of an algebraic field extension over the quotien field of \(R\). In the case \(\alpha\) NEWLINEis an anti-integral element over \(R\) we will give NEWLINEa condition for a prime ideal \(p\) of \(R\) to be NEWLINE\(pR[\alpha,~\alpha^{-1}] = R[\alpha,~\alpha^{-1}]\). NEWLINEBy making use of this we will proceed mainly with the NEWLINEstudy of flatness and faithful flatness of the extension NEWLINE\(R[\alpha,~\alpha^{-1}]/R\). Let \(\eta_{1},~\cdots,~\eta_{d}\) NEWLINEbe the coefficients of the minimal polynomial of \(\alpha\) NEWLINEover the quotient field of \(R\). Then we will also investigate NEWLINEthe extension \(R[\eta_{1},~\cdots,~\eta_{d}]/R\).