DIMENSION MODULES AND MODULAR LATTICES

A module M is called a dimension module if the Goldie (uniform) dimension satisfies the formula u(A + B) + u(A boolean AND B) = u(A) + u(B) for arbitrary submodules A, B of M. Dimension modules and related notions were studied by several authors. In this paper, we study them in a more general context of modular lattices with 0 to which the notion of dimension modules can be extended in an obvious way. Some constructions available in the lattice theory framework make it possible to identify several new aspects concerning the nature of dimension lattices and modules as well as to describe a number of related properties. In particular we find a lattice which can be used to test whether a given lattice or a module satisfies the studied properties. Most of the results are obtained for lattices and then they are applied to modules. However the examples are given, when possible, in the more restrictive case of modules.