  The classical Artin--Whaples approximation theorem allows to simultaneously
approximate finitely many different elements of a field with respect to
finitely many pairwise inequivalent absolute values. Several variants and
generalizations exist, for example for finitely many (Krull) valuations, where
one usually requires that these are independent, i.e. induce different
topologies on the field. Ribenboim proved a generalization for finitely many
valuations where the condition of independence is relaxed for a natural
compatibility condition, and Ershov proved a statement about simultaneously
approximating finitely many different elements with respect to finitely many
possibly infinite sets of pairwise independent valuations. We prove
approximation theorems for infinite sets of valuations and orderings without
requiring pairwise independence.

Commutative Algebra;Number Theory