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The defect of a one-dimensional local ring

Published online by Cambridge University Press:  26 February 2010

D. Kirby
D. Kirby
Affiliation:
The University, Leeds, 2.
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Extract

In a recent paper Northcott [3] introduced the notion of the reduction number of a one-dimensional local ring, and demonstrated its importance in the theory of abstract dilatations. In the present paper we define the reduction number of an ideal which is primary for the maximal ideal of a one-dimensional local ring, and show that under certain necessary and sufficient conditions the reduction numbers can take only a finite number of values.

Type
Research Article
Information
Mathematika , Volume 6 , Issue 2 , December 1959 , pp. 91 - 97
Copyright
Copyright © University College London 1959

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References

1.Northcott, D. G., Ideal Theory (Cambridge, 1953).Google Scholar
2.Northcott, D. G., “A general theory of one-dimensional local rings”, Proc. Glasgow Math. Soc., 2 (1956), 159–169.Google Scholar
3.Northcott, D. G., “The reduction number of a one-dimensional local ring”, Mathematika, 6 (1959), 87–90.Google Scholar
4.Northcott, D. G., and Rees, D., “A note on reductions of ideals with an application to the generalized Hilbert function”, Proc. Cambridge Phil. Soc., 50 (1954), 353–359.Google Scholar
5.Rees, D., “Two classical theorems of ideal theory”, Proc. Cambridge Phil. Soc., 52 (1956), 155–157.Google Scholar
6.Samuel, P., Algèbre Locale, Mém. Sci. Math., 123 (Paris, 1953).Google Scholar