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ON THE GROWTH OF IDEALS AND SUBMODULES

Published online by Cambridge University Press:  01 October 1997

DAN SEGAL
DAN SEGAL
Affiliation:
All Souls College, Oxford, OX1 4AL
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Abstract

All rings in this paper are commutative, with identity. If the ring R is either finitely generated, or semi-local (by which I mean Noetherian and possessing only finitely many maximal ideals), then R has only a finite number of ideals of each finite index. Thus it makes sense to study the function n[map ]an(R), where an(R) denotes the number of ideals of index n in R. In an earlier note [10], I determined an(R) for certain 2-dimensional domains R. The results to be discussed here are less precise, but more general; in particular, they deal with submodules of a finitely generated module. This makes it possible to translate them into results about the growth of normal subgroups in certain metabelian groups, the original motivation for this work: for example, we determine the finitely generated metabelian groups with ‘polynomial normal subgroup growth’ (Theorem 6.1).

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 245 - 263
Copyright
The London Mathematical Society 1997

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