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Noether Lattices Representable as Quotients of the Lattice of Monomially Generated Ideals of Polynomial Rings

Published online by Cambridge University Press:  20 November 2018

D. D. Anderson ,
E. W. Johnson  and
J. A. Johnson
D. D. Anderson
Affiliation:
The University of Iowa, Iowa City, Iowa
E. W. Johnson
Affiliation:
The University of Houston, Houston, Texas
J. A. Johnson
Affiliation:
The University of Houston, Houston, Texas
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Noether lattices were introduced by R. P. Dilworth in [5] and constitute a natural abstraction of the lattice of ideals of a Noetherian ring. In his definitive work, Dilworth showed that a minimal prime of an element generated by n principal elements has rank ≦ n. Following standard ring theoretical terminology, a local Noether lattice with (unique) maximal element M is said to be regular if M has rank n and can be generated by n principal elements.

In [3], K. P. Bogart showed that a distributive regular local Noether lattice of Krull dimension n is isomorphic to RLn, the sublattice of all ideals generated by monomials of any polynomial ring (k a field). In a later paper [4], Bogart extended his results on distributive regular local Noether lattices by showing that any distributive local Noether lattice is the image of a multiplicative map θ which preserves joins, and can in fact be thought of as the related congruence lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 789 - 799
Copyright
Copyright © Canadian Mathematical Society 1979

References

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