Approximation Theorems for Manis Valuations

Source abstract

Abstract Throughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair ( R v , P v ) of a total quotient ring T ( R ) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.