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Approximation Theorems for Manis Valuations

Published online by Cambridge University Press:  20 November 2018

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

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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

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Miroslav, Arapović, Approximation theorems for fields and commutative rings, Glasnik Mat., 18 (38) (1983), pp. 6166.Google Scholar
2. Boisen, Monte B. Jr., and Larsen, Max D., Prüfer and valuation rings with zero divisors, Pacific J. Math., 40 (1972), pp. 712.Google Scholar
3. Joachim, Gräter, Der Approximationssatz für Manisbewertungen, Arch. Math., 37 (1981), pp. 335340.Google Scholar
4. Joachim, Gräter, Der allgemeine Approximationssatz für Manisbewertungen, Mh. Math. 93 (1982), pp. 277288.Google Scholar
5. Griffin, Malcolm P., Valuations and Prufer rings, Can. J. Math. 26 (1974), pp. 412429.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. 193198.Google Scholar
8. Paulo, Ribenboim, Thorie des Valuations, University of Montreal Press, Montreal, 1964.Google Scholar