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M-Primary Elements of a Local Noether Lattice

Published online by Cambridge University Press:  20 November 2018

E. W. Johnson  and
J. A. Johnson
E. W. Johnson
Affiliation:
The University of Iowa, Iowa City, Iowa
J. A. Johnson
Affiliation:
The University of Houston, Houston, Texas
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In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 327 - 331
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Dilworth, R. P., Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498.Google Scholar
2. Johnson, E. W., A-transforms and Hilbert functions on local lattices, Trans. Amer. Math. Soc. 187 (1969), 125–139.Google Scholar