Skip to main content

Protoproducts

  • 1056 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1999)

Abstract

In Chapter 4, we used ultraproducts to derive uniform bounds for various algebraic operations, where the bounds are given in terms of the degrees of the polynomials involved. This was done by constructing a faithfully flat embedding of the polynomial ring A into an ultraproduct U(A) of polynomial rings, called its ultra-hull. Moreover, A is characterized as the subring of U(A) of all elements of finite degree. In this chapter, we want to put these uniformity results in a more general context, by replacing the degree on A by what we will call a proto-grading.

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   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aschenbrenner, M.: Ideal membership in polynomial rings over the integers. J. Amer. Math. Soc. 17(2), 407–441 (2004) 140

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Aschenbrenner, M.: Bounds and definability in polynomial rings. Quart. J. Math. 56(3), 263–300 (2005) 63, 142

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Rothmaler, P.: Introduction to model theory, Algebra, Logic and Applications, vol. 15. Gordon and Breach Science Publishers, Amsterdam (2000) 8, 11

    Google Scholar 

  4. Schoutens, H.: Absolute bounds on the number of generators of Cohen-Macaulay ideals of height at most two. Bull. Soc. Math. Belg. 13, 719–732 (2006) 118

    MATH  MathSciNet  Google Scholar 

  5. Schoutens, H.: Dimension theory for local rings of finite embedding dimension (2010). ArXiv:0809.5267v1 5, 113, 114, 118, 125, 149, 161, 166

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hans Schoutens .

Rights and permissions

Reprints and Permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Schoutens, H. (2010). Protoproducts. In: The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics(), vol 1999. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13368-8_9

Download citation