About primary lattices of a large geometric dimension (Q2386439)

For \(d \geq 3\), \textit{B. Jónsson} and \textit{G. S. Monk} [Pac. J. Math. 30, 95--139 (1969; Zbl 0186.02204)] characterized the lattices \(L(M)\) of submodules of finitetly generated \(R\)-modules \(M\), \(R\) a completely primary uniserial ring, containing a rank \(d\) free submodule: \(L\) is a primary Arguesian lattice of geometric dimension \(d\). Such an \(L\) is embeddable into the lattice \(L(V)\) of subspaces of some vector space \(V\) if and only \(R\) has prime characteristic [\textit{A. Huhn} and the reviewer, Math. Z. 144, 185--194 (1975; Zbl 0316.06006)]. Thus, the main claim of the paper under review is not upheld. The case of \(d=2\) remains open. Here, \textit{G. S. Monk} [Pac. J. Math. 30, 175--186 (1969; Zbl 0186.02301)] constructed a primary sublattice of a lattice \(L(V)\) which is not isomorphic to any \(L(M)\) and disproves the claim made by \textit{G. Takách} and the reviewer [Beitr. Algebra Geom. 46, No. 1, 215--239 (2005; Zbl 1075.06004)].