High degree anti-integral extensions of Noetherian domains

Abstract

Introduction.Let R be a Noetherian integral domain and R [X] a polynomial ring.Let a be an element of an algebraic field extension L of the quotient field K of R and let : R [X] -> R [a] be the -algebra homomorphism sending X to a. Let (X) be the monic minimal polynomial of a over K with deg (X)=d and write R[a] is called an anti-integral extension of R. In the case K(a)=K, an anti-integral elemet a is the same as an anti-integral element (i.e., R=R[a] \R[l/(X\) defied in [5].The element a is called a super-primitive element of degree d over R if J ^p for all primes p of depth one.For p^ Spec (R), k(p) denotes the residue field R p /pR p and rank^) R [a] R k(p) denotes the dimension as a vector space over k(p).We are interested in characterizing the flatness and the integrality of an anti-integral extension R\c\ of R. Indeed, among others we obtain the following results:Concerning a super-primitive element, we obtain that if R is a Krull domain and a is an algebraic element over R, then a is a super-primitive element.We also obtain that a super-primitive element is an anti-integral element.More precisely, a is super-primitive over R if and only if a is anti-integral over R and R [a] p is flat over R p for any prime ideal p of depth one.Using these results, we obtain the following: Let (5) denote the set {peSpec(jR) \rank k ( p )S(& R k(p)=d}, where S is an extension of R of degree d and let Dp^R) denote the set of all prime ideals of R of depth one.Assume that [L: K]=d, and that a y , a n ^L are anti-integral elements of degree d, and let A=R[a 1 , -, a n ].If ([J)lD<Dp(/2) (1 <i<ri)