Modules: Duals and principally generated fake duals
Let M be a module over the commutative Noetherian ring R (with identity). Observations of \textit{D. D. Anderson} [J. Algebra 47, 425-432 (1977; Zbl 0383.13011)] concerning the behaviour of the lattice dual of the submodule lattice of M lead the author, in the paper under review, to study a Noether lattice \({\mathcal L}\) in the sense of \textit{R. P. Dilworth} [Pac. J. Math. 12, 481-498 (1962; Zbl 0111.041)] and a complete modular lattice L on which there is defined a multiplication by elements of \({\mathcal L}\) satisfying certain properties. On the assumption that L is principally generated and that \({\mathcal L}\) is L-principally generated, the author investigates conditions under which the lattice dual \(L^*\) is principally generated. There are consequences in commutative algebra, including an algebraic characterization of complete local rings.
scientific article; zbMATH DE number 4061305