Compatibility of an ordering of higher level with a valuation pair for a commutative ring

The main aim of this paper is to extend the compatibility of an ordering of higher level with a valuation pair on a field to a commutative ring with unit.We introduce the concept of compatibility of an ordering of higher level with a valuation pair for a commutative ring with u-nit.Then we give four equivalent conditions for an ordering of higher level to be compatible with a valuation pair.It is a key in this paper to characterize the compatibility of an ordering of higher level with a valuation pair by means of the convexity of the valuation ring.The technique of this paper is to convert the problems in the case of rings into that of fields by taking specializations of orderings and the quotient fields of domains.