On the integral closedness of the ring \(R [\alpha] \cap R [1/ \alpha]\) (Q1293108)

Let \(R\) be an Noetherian integral domain with quotient field \(K\) and integral closure \(R'\). Let \(L\) be an algebraic field extension of \(K\), \(\alpha\in L\) and \(\varphi_{\alpha}(X)=X^d+\eta_1X^{d-1}+...+\eta_d\) the minimal polynomial of \(\alpha\) over \(K\). Consider the following ideals of \(R\), \(I_{[\alpha]}=\{a\in R;\;a\varphi_\alpha(X)\in R[X]\}\), \(J_\alpha=I_{[\alpha]}(1,\eta_1,...,\eta_d)\) and let \(\pi:R[X]\rightarrow R[\alpha]\) be the \(R\)-algebra homomorphism sending \(X\) to \(\alpha\). According to \textit{S. Oda, J. Sato} and \textit{K. Yoshida} [Osaka J. Math. 30, 119--135 (1993; Zbl 0782.13013)], \(\alpha\) is called an anti-integral element over \(R\) if \(\text{ker}(\pi)=I_{[\alpha]}\varphi_\alpha(X)R[X]\) and \(\alpha\) is called a super-primitive element over \(R\) if \(J_{[\alpha]}\not\subseteq P\) for all depth-one prime ideals \(P\) of \(R\) (it is known that a super-primitive element is anti-integral). In the paper under review, the authors continue the study of these notions. One of the results they obtain is the following. Assume that \(\alpha\) is a super-primitive element over \(R\), \(R'\) is a finitely generated \(R\)-module and the dimension formula holds between \(R\) and \(R'\). Then \(I_{[\alpha]}\) is integrally closed iff \(\text{grade}(I_{[\alpha]}+(R:_RR'))>1\). Also, \(R[\alpha]\cap R[1/\alpha]\) is integrally closed in \(R[\alpha]\) iff \(I_{[\alpha]}\) is integrally closed and either \((I_{[\alpha]})_P=PR_P\) or \((I_{[\alpha]})_P=(R:_R\eta_1)_P\) for each \(P\in \text{Ass}_R(R/I_{[\alpha]})\).