Henselian Valuations And Semiorderings On Commutative Rings

The purpose of this paper is to investigate henselian valuations on commutative rings. Some relevant results on henselian valuations of fields are generalized to the category of commutative rings. With the aid of the theory of real fields, some basic results are established. As an important result, we establish several sufficient and necessary conditions for a henselian valuation on a ring to be compatible with every semiordering (see Theorem 3.1). Besides, a counter-example is given to illustrate that an ordering need not be compatible with a henselian valuation on a ring.