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A lattice of extension rings for a commutative ring

Published online by Cambridge University Press:  24 October 2008

D. Kirby  and
M. R. Adranghi
D. Kirby
Affiliation:
University of Southampton
M. R. Adranghi
Affiliation:
University of Southampton
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Extract

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 225 - 234
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Kirby, D.The defect of a one-dimensional local ring. Mathematika 6 (1959), 91–97.Google Scholar
(2)Kirby, D.A note on superficial elements of an ideal in a local ring. Quart. J. Math. Oxford (2) 14 (1963), 21–28.Google Scholar
(3)Matlis, E.The multiplicity and reduction number of a one-dimensional local ring. Proc. London Math. Soc. 26 (1ʘ73), 273–288.Google Scholar
(4)Northcott, D. G.The neighbourhoods of a local ring. J. London Math. Soc. 30 (1955), 360–375.CrossRefGoogle Scholar
(5)Northcott, D. G.On the notion of a first neighbourhood ring with an application to the AF + BΦ theorem. Proc. Cambridge Philos. Soc. 53 (1957), 43–56.Google Scholar
(6)Northcott, D. G.The reduction number of a one-dimensional local ring. Mathematika 6 (1959), 87–90.CrossRefGoogle Scholar
(7)Zariski, O. and Samuel, P.Commutative algebra, vo1. 1 (Princeton, van Nostrand, 1958).Google Scholar