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A Structural Approach to Noether Lattices

Published online by Cambridge University Press:  20 November 2018

E. W. Johnson ,
J. A. Johnson  and
J. P. Lediaev
E. W. Johnson
Affiliation:
University of Iowa, Iowa City, Iowa
J. A. Johnson
Affiliation:
University of Houston, Houston, Texas
J. P. Lediaev
Affiliation:
University of Iowa, Iowa City, Iowa
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In this paper we explore the extent to which embedding and isomorphism questions about a Noether lattice ℒ can be reduced to questions about simpler structures associated with ℒ.

In § 1, we use a variation of Dilworth's congruence approach [2] to associate a collection of semi-local Noether lattices with a given Noether lattice ℒ. We show that these semi-localizations determine ℒ to within isomorphism (Corollary 1.5); thus embedding and isomorphism questions about ℒ are largely reduced to the semi-local case.

In § 2, we consider the influence on a semi-local Noether lattice ℒ of the substructure ∂ ℒ consisting of all elements, all of whose associated primes are maximal. Here we find that if ∂ ℒ can be embedded in a semi-local Noether lattice ℒ*, then ℒ can be embedded in an extension of ℒ*.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 657 - 665
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bogart, K. P., Structure theorems for regular local Noether lattices, Michigan Math. J. 15 (1968), 167–176.Google Scholar
2. Dilworth, R. P., Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498.Google Scholar
3. Johnson, E. W., A-transforms and Hilbert functions in local lattices, Trans. Amer. Math. Soc. 137 (1969), 125–140.Google Scholar
4. Johnson, E. W. and Johnson, J. A., M-primary elements of a local Noether lattice, Can. J. Math. 22 (1970), 327–331.Google Scholar
5. Zariski, O. and Samuel, P., Commutative algebra, Vol. II, The University Series in Higher Mathematics (Van Nostrand, Princeton, N.J.-Toronto-London-New York, 1960).Google Scholar