ABSTRACT
Let R be a Noetherian domain and R[X ] a polynomial ring. Let α be an element of an algebraic field extension L of the quotient field K of R and π : R[X ] → R[α] be the R-algebra homomorphism sending X to α. Put E (α) := Ker(π ), an ideal of R. Let ϕα(X ) be the monic minimal polynomial of α over K with deg ϕα(X ) = d and write
ϕα(X ) = Xd + η1 Xd−1 + · · · + ηd
Then ηi (1 ≤ i ≤ d) are uniquely determined by α. Let Iηi := R :R ηi and ηi :=
⋂d i=1 Iηi , the latter of which is called a generalized denominator ideal of
α. We say that α is an anti-integral element if and only if E (α) = I[α]ϕα(X )R[X ]. For f (X ) ∈ R[X ], let c( f (X )) denote the ideal of R generated by the coefficients of f (X ). For an ideal J of R[X ], let c(J ) denote the ideal generated by the coefficients of the elements in J . If α is an anti-integral element, then c(E (α)) = c(I[α]ϕα(X )R[X ]) = I[α](1, η1, . . ., ηd ). Put J[α] = I[α](1, η1, . . ., ηd ). If J[α] ⊆ p for all p ∈ Dp1(R) := {p ∈ Spec(R)|depth(Rp) = 1}, then α is called a super-primitive element. It is known that a super-primitive element is an anti-integral element (cf. Theorem 2.2.8). By definition, the super-primitive is characterized by the set of Dp1(R). Put R〈α〉 := R[α] ∩ R[α−1].
