Sublattices of regular elements
 D. D. Anderson,
 E. W. Johnson,
 Richard L. Spellerberg II
 … show all 3 hide
Abstract
Let L be an rlattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: \(L_f = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right\left. {(0:A) = 0} \right\},L_{sr} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right\) there is a compact element \(X \leqslant A\) with \(\left. {(0:X) = 0} \right\},L_r = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right\) there is a principal element \(X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \rightA = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha \) is a principal element with (0:X_{\alpha })=0\} . The first three subsets L_{f}, L_{sr}, and L_{r} are augmented filters \(\mathcal{L}^0 \) on L, i.e., \(\mathcal{L}^0 = \mathcal{L} \cup \left\{ 0 \right\}{\text{ where }}\mathcal{L}\) is a multiplicatively closed subset of $L$ with $A\in \(\mathcal{L}\) and $B\geq A$ with $B\in L$ implies $B\in \(\mathcal{L}\) and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$lattice or Noether lattice (i.e., an $r$lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$module.
 D. D. Anderson, Distributive Noether lattices, Michigan Math. J. 22 (1975), 109–115.
 D. D. Anderson, Abstract commutative ideal theory without chain condition, Algebra Universalis 6 (1976), 131–145.
 D. D. Anderson, Fake rings, fake modules, and duality, J. Algebra 47 (1977), 425–432.
 D. D. Anderson and E. W. Johnson, Ideal theory in commutative semigroups, SemigroupForum 30 (1984), 127–158.
 D. D. Anderson and E. W. Johnson, Dilworth's principal elements, Algebra Universalis 36 (1996), 392–404.
 D. D. Anderson and J. Pascual, Regular ideals in commutative rings, sublattices of regular ideals, and Prüfer rings, J. Algebra 111 (1987), 404–426.
 K. P. Bogart, Structure theorems for regular local Noether lattices, Michigan Math. J. 15 (1968), 167–176.
 K. P. Bogart, Distributive local Noether lattices, Michigan Math. J. 16 (1969).
 R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498.
 E. W. Johnson, Atransforms of Noether lattices, Dissertation, University of California, Riverside, 1966.
 E. W. Johnson and J. P. Lediaev, Join principal elements in Noether lattices, Proc. Amer. Math. Soc. 36 (1972), 73–78.
 I. Kaplansky, Commutative Rings, revised ed., Polygonal Publishing House, Washington, N.J., 1994.
 R. L. Spellerberg II, Some problems in multiplicative lattice theory, Dissertation, The University of Iowa, 1990.
 Title
 Sublattices of regular elements
 Journal

Periodica Mathematica Hungarica
Volume 44, Issue 1 , pp 111126
 Cover Date
 20020301
 DOI
 10.1023/A:1014932204184
 Print ISSN
 00315303
 Online ISSN
 15882829
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 D. D. Anderson ^{(1)}
 E. W. Johnson ^{(1)}
 Richard L. Spellerberg II ^{(2)}
 Author Affiliations

 1. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, U.S.A.
 2. Department of Mathematics, Simpson College, Indianola, IA, 50125, U.S.A.