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Semi-regular local rings

Published online by Cambridge University Press:  26 February 2010

D. G. Northcott
D. G. Northcott
Affiliation:
The University, Sheffield
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Extract

In the following pages there will be found an account of the properties of a certain class of local rings which are here termed semi-regular local rings. As this name will suggest, these rings share many properties in common with the more familiar regular local rings, but they form a larger class and the characteristic properties are preserved under a greater variety of transformations. The first occasion on which these rings were studied by the author was in connection with a problem concerning the irreducibility of certain ideals, but about the same time they were investigated in much greater detail by Rees [7] and in quite a different connection. In his discussion, Rees made considerable use of the ideas and techniques of homological algebra. Here a number of the same results, as well as some additional ones, are established by quite different methods. The essential tools used on this occasion are the results obtained by Lech [3] in his important researches concerning the associativity formula for multiplicities. Before describing these, we shall first introduce some notation which will be used consistently throughout the rest of the paper.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 2 , December 1956 , pp. 117 - 126
Copyright
Copyright © University College London 1956

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References

1.Cohen, I. S., “On the structure and ideal theory of complete local rings”, Trans. American Math. Soc., 59 (1946), 54–106.CrossRefGoogle Scholar
2.Krull, W., “Dimensionstheorie in Stellenringen”, J. reine angew. Math., 179 (1938), 204–226.CrossRefGoogle Scholar
3.Lech, C., “On the associativity formula for multiplicities”, Arkiv för Matematik (1956), 301–314.Google Scholar
4.Northcott, D. G., Ideal Theory (Cambridge Tracts No. 42, 1953).CrossRefGoogle Scholar
5.Northcott, D. G., “Irreducible ideals in local rings”, Journal London Math. Soc. (to appear).Google Scholar
6.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.CrossRefGoogle Scholar
7.Rees, D., “The grade of an ideal or module” (to appear).Google Scholar
8.Samuel, P., Algèbre Locale, Mémorial Sci. Math., 123 (Paris 1953).Google Scholar