Maximal non valuation domains in an integral domain

Abstract

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S, and for any ring T such that R T S, T is a valuation subring of S. For a local domain S, the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim(R,S) and the number of rings between R and S is given when R is a maximal non VD in S and dim(R, S) is finite. For a maximal non VD R in S such that R R′S S and dim(R, S) is finite, the equality of dim(R,S) and dim(R′S , S) is established.