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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

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