The Semilattice of Annihilator Classes in a Reduced Commutative Ring |
| |
| Authors: | David F Anderson John D LaGrange |
| |
| Institution: | 1. Mathematics Department , University of Tennessee , Knoxville , Tennessee , USA anderson@math.utk.edu;3. Division of Natural and Behavioral Sciences , Lindsey Wilson College , Columbia , Kentucky , USA |
| |
| Abstract: | Let R be a reduced commutative ring with 1 ≠ 0. Let R E be the set of equivalence classes for the equivalence relation on R given by x ～ y if and only if ann R (x) = ann R (y). Then R E is a (meet) semilattice with respect to the order x] ≤ y] if and only if ann R (y) ? ann R (x). In this paper, we investigate when R E is a lattice and relate this to when R is weakly complemented or satisfies the annihilator condition. We also consider when R is a (meet) semilattice with respect to the Abian order defined by x ≤ y if and only if xy = x 2. |
| |
| Keywords: | Annihilator condition Lattice Reduced ring Semilattice Weakly complemented ring |
|
|
