In an earlier work of B.
Jónsson and the author it was shown that an Arguesian primary lattice of geometric
dimension at least 3 can be represented as the submodule lattice of a finitely
generated module over a completely primary uniserial ring. Inasmuch as the class of
primary lattices includes the class of subspace lattices of (nondegenerate)
projective geometries, two questions then naturally arise: (1) Is a primary
lattice of geometric dimension at least 4 Arguesian? (2) Is an Arguesian
primary lattice of geometric dimension 2 representable ? The first question is
answered in the affirmative in §1, thus showing that the abovementioned paper
subsumes the results of E. Inaba on the representation of primary lattices of
geometric dimension at least 4. A counter example is given in §2 showing
that an Arguesian lattice of geometric dimension 2 cannot, in general, be
represented, but for reasons far deeper than the cardinality arguments given for the
representability or nonrepresentability of subspace lattices of l-dimensional projective
geometries.
