On Conditions for Denominator Ideals to Diffuse and Conditions for Elements to Be Exclusive in Anti-Integral Extensions

Notation and Convensions Throughout this paper, we use the following notation unless otherwise specified: Let R be a Noetherian domain (which is commutative and has a unit), let R[X]be a polynomial ring,let. α be an element of an algebraic extension field 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 ψα(X)=Xd+η1Xd-1+...+ηd. Then ηi ∈ K (1≦i≦d) are uniquely determined by α. Put d=[K(α):K],