On some results deduced from the equality \(R[1/\alpha] =R[1/(\alpha -a)]\) (Q2709724)

An element \(\alpha\) of an algebraic extension of the quotient field of an integral domain \(R\) is called anti-integral over \(R\), if the kernel of the canonical epimorphism \(\pi\colon R[x]\rightarrow R[\alpha]\) equals \(I_{[\alpha]}\varphi_\alpha R[x]\), where \(\varphi_\alpha\) is the minimal polynomial of \(\alpha\) over \(K\) and \(I_{[\alpha]}\) is the intersection of all \((R\colon_R \eta)\), where \(\eta\) runs through the coefficients of \(\varphi_\alpha\). NEWLINENEWLINENEWLINELet \(\alpha\neq 0\) be anti-integral over a Noetherian integral domain \(R\) and \(a\in R\), \(a\neq \alpha\). The authors derive several consequences of \(R[1/\alpha]=R[1/(\alpha-a)]\), such as the existence of a unit \(\gamma\) of \(R[1/\alpha]\) for which \(1/\alpha=\gamma/(\alpha-a)\). Moreover, they show, in the following list of statementsm the equivalence of (1) and (2) and the equivalence of (3) and (4) and the implication \(2\Rightarrow 3\). NEWLINENEWLINENEWLINE(1) \(R[\alpha,{1\over \alpha}] = R[\alpha,{1\over {(\alpha-a)} }] \) NEWLINENEWLINENEWLINE(2) \(\sqrt{\alpha R[\alpha]}=\sqrt{(\alpha - a) R[\alpha]}\) NEWLINENEWLINENEWLINE(3) \(\sqrt{\alpha R[\alpha]}\supseteq \sqrt{a R[\alpha]}\) NEWLINENEWLINENEWLINE(4) \(a\in \sqrt{ I_{[\alpha^{-1}]} }\).