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Compatibility of quasi-orderings and valuations: a Baer-Krull theorem for quasi-ordered rings. (English) Zbl 1476.06009

In 1969, M. E. Manis introduced valuations on commutative rings [Proc. Am. Math. Soc. 20, 193–198 (1969; Zbl 0179.34201)], recently the class of totally quasi-ordered rings was recently developed. In this paper, given a quasi-ordered ring \((R,\preceq)\) and a valuation \(v\) on \(R\), the authors establish the notion of compatibility between \(v\) and \(\preceq\), leading to a definition of the rank of \((R,\preceq)\). The main result is a Baer-Krull theorem for quasi-ordered rings: fixing a Manis valuation \(v\) on \(R\), the authors characterize all \(v\)-compatible quasi-orders of R by lifting the quasi-orders from the residue class domain to \(R\) itself. In particular, this approach generalizes to the ring case the results of [S. Kuhlmann and G. Lehéricy, Order 35, No. 2, 283–291 (2018; Zbl 1428.06003)].

MSC:

06F25 Ordered rings, algebras, modules
13A18 Valuations and their generalizations for commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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References:

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