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An embedding theorem for HCF-rings

Published online by Cambridge University Press:  24 October 2008

Robert Gilmer
Robert Gilmer
Affiliation:
Florida State University, Tallahassee
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Extract

Let D be an integral domain with identity. In (3), Cohn has shown that if D is a Schreier ring, then D can be inertly embedded in a pre-Bezout domain B(D), which Cohn calls the pre-Bezout hull of D. Further, D is an HCF-ring if and only if B(D) is a Bezout domain, and D is a UFD if and only if B(D) is a PID. Using a modification of Cohn's techniques, Samuel in (8) has shown that a Krull domain can be inertly embedded in a Dedekind domain. In this paper, we show that these embedding theorems are also obtained by considering the embedding of a v-ring in its Kronecker function ring with respect to the v-operation. To wit, if D is a v-ring and if Dv is the Kronecker function ring of D with respect to the v-operation, then Dv is a Bezout domain, and we show that D is inertly embedded in Dv if an only if D is an HCF-ring. Moreover, D is a UFD if and only if Dv is a PID, and if D is a Krull domain, D is inertly embedded (in Samuel's sense) in Dv, a Dedekind domain.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 3 , November 1970 , pp. 583 - 587
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Boccioni, D.Alcune osservazioni sugli anelli pseudo-bezoutiani e fattoriali. Rend. Sem. Mat. Univ. Padova 37 (1967), 273–288.Google Scholar
(2)Bourbaki, N.Algèbre commutative, ch. 7 (Diviseurs) (Paris, 1965).Google Scholar
(3)Cohn, P. M.Bezout rings and their subrings. Proc. Cambridge Philos. Soc. 64 (1968), 251–264.Google Scholar
(4)Gilmer, R.Multiplicative ideal theory (Kingston, Ontario, 1968).Google Scholar
(5)Krull, W.Beiträge zur Arithmetik kommutativer Integritätsbereiche. Math. Z. 41 (1936), 545–577.CrossRefGoogle Scholar
(6)Krull, W.Beiträge zur Arithmetik kommutativer Integritätsbereiche II. Math. Z. 41 (1936), 665–679.CrossRefGoogle Scholar
(7)Prüfer, H.Untersuchungen über Teilbarkeitseigenschaften in Korpern. J. Heine Angew. Math, 168 (1932), 1–36.Google Scholar
(8)Samuel, P.On a construction due to P. M. Cohn. Proc. Cambridge Philos. Soc. 64 (1968), 249–250.Google Scholar
(9)van der Waekden, B. L.Modern algebra, vol. 2 (New York, 1950).Google Scholar