Skip to main content

Approximation Theorems

  • 637 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2103)

Abstract

We embed the important work of Gräter on approximation theorems in the book. Approximation theorems are a well-known and important topic in classical valuation theory of fields. The question is to decide for given valuations v 1, , v n of a field, elements a 1, , a n in the field and α 1, , α n in the value groups whether there is an element x in the field such that

$$\displaystyle{v_{i}(x - a_{i}) \geq \alpha _{i}\mbox{ resp. }v_{i}(x - a_{i}) =\alpha _{i}}$$

for all i; i.e. if the elements a i can be approximated by some x up to a certain degree. Gräter elaborated various approximation theorems in our general setting of R-Prüfer rings and has found deep connections, to be reflected below.

Keywords

  • General Approximation Theorem
  • Classical Valuation Theory
  • Krull Type
  • Value Management
  • Convex Subgroup

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    This means that f(α) ≥ f(β) if α ≥ β (cf. [Vol. I, p. 17]). Note that necessarily \(f(\varGamma _{v}) \subset \varGamma _{w}\) and that \(f\vert \varGamma _{v}:\varGamma _{v} \rightarrow \varGamma _{w}\) is a homomorphism of ordered groups.

  2. 2.

    Note that then necessarily f() = .

References

  1. J. Alajbegović, Approximation theorems for Manis valuations with the inverse property. Commun. Algebra 12, 1399–1417 (1984)

    CrossRef  MATH  Google Scholar 

  2. J. Alajbegović, R-Prüfer rings and approximation theorems, in Methods in Ring Theory, ed. by F. van Ostayen (D. Reidel Publisher, Dordrecht/Boston/London, 1984), pp. 1–36

    CrossRef  Google Scholar 

  3. J. Alajbegović, J. Močkoř, Approximation Theorems in Commutative Algebra (Kluwer Academic, Dordrecht, 1992)

    CrossRef  MATH  Google Scholar 

  4. J. Alajbegovic, E. Osmanagić, Essential valuations of Krull rings with zero divisors. Commun. Algebra 18, 2007–2020 (1990)

    CrossRef  MATH  Google Scholar 

  5. M. Arapović, Approximation theorems for Manis valuations. Can. Math. Bull. 28(2), 184–189 (1985)

    CrossRef  MATH  Google Scholar 

  6. O. Endler, Valuation Theory (Springer, Berlin, 1972)

    CrossRef  MATH  Google Scholar 

  7. J. Gräter, Der allgemeine Approximationssatz für Manisbewertungen. Mh. Math. 93, 277–288 (1982)

    CrossRef  MATH  Google Scholar 

  8. J. Gräter, Der Approximationssatz für Manisbewertungen. Arch. Math. 37, 335–340 (1981)

    CrossRef  MATH  Google Scholar 

  9. J. Gräter, R-Prüferringe und Bewertungsfamilien. Arch. Math. 41, 319–327 (1983)

    CrossRef  MATH  Google Scholar 

  10. M. Griffin, Rings of Krull type. J. Reine Angew. Math. 229, 1–27 (1968)

    MATH  MathSciNet  Google Scholar 

  11. M. Griffin, Families of finite character and essential valuations. Trans. Am. Math. Soc. 130, 75–85 (1968)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. M. Larsen, P. McCarthy, Multiplicative Theory of Ideals (Academic, New York, 1971)

    MATH  Google Scholar 

  13. M.E. Manis, Valuations on a commutative ring. Proc. Am. Math. Soc. 20, 193–198 (1969)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. P. Ribenboim, The Theory of Classical Valuations (Springer, Berlin, 1999)

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Knebusch, M., Kaiser, T. (2014). Approximation Theorems. In: Manis Valuations and Prüfer Extensions II. Lecture Notes in Mathematics, vol 2103. Springer, Cham. https://doi.org/10.1007/978-3-319-03212-2_2

Download citation