Dimension de l’anneau des polynomes a valeurs entieres

Abstract

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.