On the coordinatization of primary Arguesian lattices of low geometric dimension

Abstract

Correcting claims made in Herrmann and Takach (Beitr Algebr Geom 46:215–239, 2005), we give lattice theoretic characterizations of lattices, \(L\), isomorphic to submodule lattices of finitely generated modules over commutative completely primary uniserial rings and of those isomorphic to subgroup lattices of finite abelian \(p\)-groups. Dealing with coordinatization over arbitrary completely primary uniserial rings, we have to exclude the case that \(L\) has breadth \(\ge 3\) and all but \(2\) basis elements are atoms. Primary Arguesian lattices \(L\) of the latter type are shown to admit a cover preserving embedding into the subspace lattice of some vector space. The approach is that of Herrmann and Takach (Beitr Algebr Geom 46:215–239, 2005) but takes into account Monk’s construction of non-coordinatizable primary Arguesian lattices of the exceptional types.