Local properties on anti-integral elements (Q2900200)

Let \(R\) be a Noetherian integral domain with quotient field \(K\), and let \(K(\alpha)\) be an algebraic field extension of \(K\) of degree \(d\). Let \(\varphi_{\alpha}(X) = X^d + a_1X^{d-1} + \cdots + a_d\) be the minimal polynomial of \(\alpha\) over \(K\), \(I_{[\alpha]} = (R:_Ra_1) \cap \cdots \cap (R:_Ra_d)\), and \(\pi : R[X] \longrightarrow R[\alpha]\) be the \(R\)-algebra homomorphism sending \(X\) to \(\alpha\). Then \(\alpha\) is called an anti-integral element of degree \(d\) over \(R\) if Ker\(\pi = I_{[\alpha]}\varphi_{\alpha}(X)R[X]\). Among other things, the authors show that if if \(R_P[\alpha]\) is flat and integral over \(R_P\) for every depth-one prime ideal \(P\) of \(R\), and \([K(\beta) : K] = d\) for \(\beta \in R[\alpha]\), then \(\beta\) is an anti-integral element of degree \(d\) over \(R\).