Self-duality and co-multiplication lattices (Q1119671)

The notation and terminology is from \textit{D. D. Anderson} [ibid. 6, 131- 145 (1976; Zbl 0355.06022); J. Algebra 47, 425-432 (1977; Zbl 0383.13011)]. If L is a fake module over the multiplicative lattice \({\mathcal L}\), then its lattice dual is a fake module over \({\mathcal L}\) with product \(A*N=N:A\). In this paper two main properties of elements are studied: P: \(A\leq E\) implies \(A=EC\) for some C, and an apparently dual property \(P^*:\) \(A\geq E\) implies \(A=E:C\) for some C. An element \(B\in L\) is a (co- )multiplication element if it satisfies P as an element of L \((L^*)\). L is a (generalized) P-lattice or a (generalized) multiplication lattice if every element (\(\neq 1)\) is a multiplication, and (generalized) \(P^*\)- lattice or co-multiplication lattice if every element (\(\neq 0)\) is co- multiplication. In this paper the author obtains, using the dual nature of the properties P and \(P^*\), the structure of a multiplicative lattice \({\mathcal L}\) satisfying \(P^*\) identically: a Noether PE-lattice of Krull dimension 0. This is then applied to obtain the structure of a multiplicative lattice in which the nonzero elements satisfy \(P^*\). This allows to determine precisely when such a lattice is Noether and to give precise conditions under which \({\mathcal L}\) is the lattice of ideals of a Noetherian ring.