Valuations and convex subrings of a commutative ring with higher level preordering

The purpose of this paper is to investigate general valuations and convex subrings of a commutative ring with higher level preordering. By defining the compatibility between valuations and preorderings (or orderings) of higher level in a commutative ring, we discuss the interplay between valuations and orderings of higher level. It is shown that the convex subrings are exactly valuation subrings compatible with some ordering. Moreover, the elements in the convex hull of a subring with respect to a given preordering are described. In this paper, the emphasis on Manis valuations is removed.