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)].


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


[1] Bourbaki, N.: Algèbre commutative, Chap. 1-7, Hermann Paris, 1961-1965
[2] Engler, A.J., Prestel, A.: Valued fields springer monographs in mathematics (2005) · Zbl 1128.12009
[3] Fakhruddin, S.M.: Quasi-ordered fields. J. Pure Appl. Algebr. 45, 207-210 (1987) · Zbl 0629.12022 · doi:10.1016/0022-4049(87)90069-7
[4] Griffin, M.: Valuations and Prüfer rings. Can. J. Math. XXVI(2), 412-429 (1974) · Zbl 0259.13008 · doi:10.4153/CJM-1974-042-1
[5] Knebusch, M., Kaiser, T.: Manis Valuations and Prüfer Extensions II, vol. 2103. Springer LNM (2014) · Zbl 1296.13001
[6] Knebusch, M., Zhang, D: Manis Valuations and Prüfer Extensions I - A New Chapter in Commutative Algebra, vol. 1791. Springer LNM (2002) · Zbl 1033.13001
[7] Kuhlmann, S., Lehéricy, G.: A Baer-Krull Theorem for Quasi-Ordered Groups Order. https://doi.org/10.1007/s11083-017-9432-5 (2017) · Zbl 1428.06003
[8] Kuhlmann, S., Matusinski, M., Point, F: The valuation difference rank of a quasi-ordered difference field. Groups, Modules and Model Theory - Surveys and Recent Developments in Memory of Rüdiger Göbel, pp. 399-414. Springer (2017) · Zbl 1436.03202
[9] Lam, T.Y.: Orderings, valuations and quadratic forms. Amer. Math. Soc. Regional Conference Series in Math. vol. 52. Providence (1983) · Zbl 0516.12001
[10] Manis, E.M.: Valuations on a commutative ring. Proc. Amer. Math. Soc. 20, 193-198 (1969) · Zbl 0179.34201 · doi:10.1090/S0002-9939-1969-0233813-2
[11] Müller, S.: Quasi-ordered rings. Commun. Algebra 46(11), 4979-4985 (2017). arXiv:1706.04533
[12] Powers, V.: Valuations and higher level orders in commutative rings. J. Algebra 172, 255-272 (1995) · Zbl 0845.11016 · doi:10.1016/S0021-8693(05)80002-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.