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Some rings of interest in the study of places

Published online by Cambridge University Press:  24 October 2008

J. T. Knight
J. T. Knight
Affiliation:
Churchill College, Cambridge
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Extract

The rings studied in this paper were first constructed in connexion with the problem (Proposition 1) of extending a place from an integral domain to its field of fractions. They provide (Proposition 3) a series of examples of places not so extendable, generalizing an example implicit in the work of Samuel(3); and by the use of ultraproducts they show that such extendability is (in certain precise sense) not an elementary property (Proposition 4). On the other hand they have some unexpected properties, especially in connexion with the relation 1/y0 + … + 1/yr = 1, and they have very small automorphism groups (Proposition 2 and Corollaries); and they are not unique factorization rings (Proposition 5), though they are integrally closed (Proposition 6).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 2 , September 1970 , pp. 255 - 264
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Chevalley, C.Algebraic functions of one variable, p. 6, (American Mathematical Society 1951).CrossRefGoogle Scholar
(2)Kochen, S.Ultraproducts in the theory of models. Ann. of Math. (2) 74 (1961), 221–261.CrossRefGoogle Scholar
(3)Samuel, P.La notion de place dans un anneau. Bull. Soc. Math. France 85 (1957), 123–133.CrossRefGoogle Scholar