## Abstract

Let L be an r-lattice, 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 \(X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|A = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha \) 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.

## Preview

Unable to display preview. Download preview PDF.

### REFERENCES

- [1]
- [2]D. D. Anderson, Abstract commutative ideal theory without chain condition,
*Algebra Universalis***6**(1976), 131–145.Google Scholar - [3]
- [4]D. D. Anderson and E. W. Johnson, Ideal theory in commutative semigroups,
*SemigroupForum***30**(1984), 127–158.Google Scholar - [5]D. D. Anderson and E. W. Johnson, Dilworth's principal elements,
*Algebra Universalis***36**(1996), 392–404.Google Scholar - [6]D. D. Anderson and J. Pascual, Regular ideals in commutative rings, sublattices of regular ideals, and Prüfer rings,
*J. Algebra***111**(1987), 404–426.Google Scholar - [7]K. P. Bogart, Structure theorems for regular local Noether lattices,
*Michigan Math. J.***15**(1968), 167–176.Google Scholar - [8]
- [9]R. P. Dilworth, Abstract commutative ideal theory,
*Pacific J. Math.***12**(1962), 481–498.Google Scholar - [10]E. W. Johnson,
*A*-*transforms of Noether lattices*, Dissertation, University of California, Riverside, 1966.Google Scholar - [11]E. W. Johnson and J. P. Lediaev, Join principal elements in Noether lattices,
*Proc. Amer. Math. Soc.***36**(1972), 73–78.Google Scholar - [12]I. Kaplansky,
*Commutative Rings*, revised ed., Polygonal Publishing House, Washington, N.J., 1994.Google Scholar - [13]R. L. Spellerberg II,
*Some problems in multiplicative lattice theory*, Dissertation, The University of Iowa, 1990.Google Scholar