On real closed rings of higher level

The purpose of this paper is to introduce the notion of real closed rings of higher level in the category of reduced commutative rings. Such real closed rings of higher level are just generalized real closed fields in case the rings considered are fields. With the aid of the theory of totally integrally closed rings, some important results on real closed rings of higher level are established. The results in this paper involve real valuations and a related characterization of real closed rings of higher level.