Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T18:27:44.738Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation Theorems for Manis Valuations

Published online by Cambridge University Press:  20 November 2018

Miroslav Arapović
Miroslav Arapović*
Affiliation:
Department of mathematics, University of sarajevo71000 Sarajevo, Yugoslavia
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.

Throughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair (Rv, Pv) of a total quotient ring T(R) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.

Keywords

13A1813F05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 2 , 01 June 1985 , pp. 184 - 189
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Miroslav, Arapović, Approximation theorems for fields and commutative rings, Glasnik Mat., 18 (38) (1983), pp. 61–66.Google Scholar
2. Boisen, Monte B. Jr., and Larsen, Max D., Prüfer and valuation rings with zero divisors, Pacific J. Math., 40 (1972), pp. 7–12.Google Scholar
3. Joachim, Gräter, Der Approximationssatz für Manisbewertungen, Arch. Math., 37 (1981), pp. 335–340.Google Scholar
4. Joachim, Gräter, Der allgemeine Approximationssatz für Manisbewertungen, Mh. Math. 93 (1982), pp. 277–288.Google Scholar
5. Griffin, Malcolm P., Valuations and Prufer rings, Can. J. Math. 26 (1974), pp. 412–429.Google Scholar
6. Larsen, M.D. and McCarthy, P.J., Multiplicative theory of ideals, Academic Press, New York, 1971.Google Scholar
7. Manis, Marie E., Valuations on a commutative ring, Proc. Amer. Math. Soc., 20 (1969), pp. 193–198.Google Scholar
8. Paulo, Ribenboim, Thorie des Valuations, University of Montreal Press, Montreal, 1964.Google Scholar