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On the Ideal Theory of the KroneckerFunction Ring and the Domain D(X)

Published online by Cambridge University Press:  20 November 2018

Jimmy T. Arnold
Jimmy T. Arnold*
Affiliation:
Florida State University, Tallahassee, Florida
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Let D be an integrally closed domain with identity having quotient field L. If {Vα} is the set of valuation overrings of D and if A is an ideal of D, then à = ∪ αAVα is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Ãf ⊆ Ag} (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prüfer domain, a Dedekind domain, or an almost Dedekind domain.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 558 - 563
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Arnold, J. T., On the dimension theory of overrings of an integral domain (submitted for publication).Google Scholar
2. Gilmer, R. W., Domains in which valuation ideals are prime powers, Arch. Math. 17 (1966), 210–215.Google Scholar
3. Gilmer, R. W., Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964), 813–818.Google Scholar
4. Krull, W., Beitrdge Zur Arithmetik kommutativer Integritdtsbereiche, Math. Z. 41 (1936), 545–577.Google Scholar
5. Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar
6. Priifer, H., Untersuchungen iiber die Teilbarkeitseigenschaften in Kôrpern, J. Reine Angew. Math. 168 (1932), 1–36.Google Scholar
7. Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (Van Nostrand, Princeton, 1958).Google Scholar
8. Zariski, O. and Samuel, P., Commutative algebra, Vol. 2 (Van Nostrand, Princeton, 1960).Google Scholar