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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

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