Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-05T09:24:44.961Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Anderson, D., Distributive Noether lattices, Michigan Math. J.. 22 (1975), 109–115.Google Scholar
2. Anderson, D., Abstract commutative ideal theory without chain condition, Algebra Universalis, 6 (1976), 131–145.Google Scholar
3. Bogart, K. P., Structure theorems for regular local Noether lattices, Michigan Math. J.. 15 (1968), 167–176.Google Scholar
4. Bogart, K. P., Distributive local Noether lattices, Michigan Math. J.. 16 (1969), 215–223.Google Scholar
5. Dilworth, R. P., Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498.Google Scholar
6. Johnson, E. W. and Detlefsen, M., Prime sequences and distributivity in local Noether lattices, Fund. Math.. 86 (1974), 149–156.Google Scholar
7. McCarthy, P. J., Homomorphisms of certain commutative lattice ordered semigroups, Acta Sci. Math. XXVII (1966), 63–65.Google Scholar