Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T21:55:00.723Z Has data issue: false hasContentIssue false
  • English
  • Français

The transcendence degree of an integral domain over a subfield and the dimension of the domain

Published online by Cambridge University Press:  22 January 2016

Hiroshi Tanimoto
Hiroshi Tanimoto*
Affiliation:
Department of Mathematics, Faculty of Education and Culture, Miyazaki University, Miyazaki 889-2192, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a domain A containing a field k with tr.degkA < ∞, we define a new transcendence degree of A with respect to k, which is denoted by tdkA. By using this, we generalize the theorem that for every affine domain A over a field k it holds that dim A = tr.degkA. For example, we show that if A is a quasi-local domain containing a field k with dim A = tdkA < ∞, then for every Noetherian local k-subalgebra R of A it holds that dim R = tdkR. Moreover we also generalize the theorem due to Gilmer, Nashier and Nichols.

Keywords

13C1513E0513B30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 170 , 2003 , pp. 145 - 162
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[1] Alamelu, S., Subrings of affine rings, J. Indian Math. Soc., 42 (1978), 203–214.Google Scholar
[2] Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993.Google Scholar
[3] Gilmer, R., Nashier, B. and Nichols, W., The prime spectra of subalgebras of affine algebras and their localizations, J. Pure Appl. Algebra, 57 (1989), 47–65.Google Scholar
[4] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math., 96 (1972), 318–337.Google Scholar
[5] Matsumura, H., Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.Google Scholar
[6] Nagata, M., Local Rings, Interscience, New York, 1962.Google Scholar
[7] Onoda, N. and Yoshida, K., On noetherian subrings of an affine domain, Hiroshima Math. J., 12 (1982), 377–384.Google Scholar