The notion of ordering of higher level was first introduced and studied by E. Becker [Hereditarily Pythagorean fields and orderings of higher level. Monogr. Mat. 29 (1978; Zbl 0509.12020)] for fields. In this theory, the role played by sums of squares in a field is replaced by that of sums of powers of higher degree, and E. Becker has obtained a successful generalization of Artin-Schreier theory. Later the theory was extended to rings [S. M. Barton, Can. J. Math. 44, No. 3, 449-462 (1992; Zbl 0766.13005); E. Becker and D. Gondard, Manuscr. Math. 65, No. 1, 63-82 (1989; Zbl 0689.12016); and M. Marshall and V. Powers, Commun. Algebra 21, No. 11, 4083-4102 (1993; Zbl 0794.11020)].
This paper investigates general valuations and convex subrings of a commutative ring with an identity element and with a higher level preordering. The paper involves arbitrary valuations, called paravaluations by J. A. Huckaba in [Commutative rings with zero divisors. Marcel Dekker (1988; Zbl 0637.13001)], instead of Manis valuations.
It is shown that the convex subrings are exactly valuation subrings “compatible” with some ordering. The author also gives the description of the elements in the convex hull of subrings with respect to a given preordering.
The article studies first convex subrings and then describes carefully orderings of exact level \(n\) in \(\mathbb{Q} (X)\). The important part 3 is devoted to compatible valuations and the last one to the description of elements in convex subrings.
All proofs are included and detailed.