Some Relationship between Anti-Integral Extensions of Noetherian Domains

Let R be a Noetherian domain with quotient field K and let α be an anti-integral element of degree d over R. Let β be an elemen of R(α) (resp. R(α,α-1)) such that β is an anti-integral element over R and that R(α) (resp. R(α,α-1)) is integral over R(β)). We shall investigate some properties descending from R(α) (resp. R(α,α-1)) to R(β), i. e., flatness and faithful flatness, and study the ideals J(α), J(β), J(α) and J(β). Let R be a Noetherian domain and R(X) a polynomial ring. Let α be an element of an algebraic extension field L of the quotient field K of R and let π: R(X) →R(α) be the R-algebra homomorphism, sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write