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].