On lattices, modules and groups with many uniform elements

The uniform dimension, also known as Goldie dimension, can be defined and used not only in the class of modules, but also in large classes of lattices and groups. For considering this dimension it is necessary to involve uniform elements.
In this paper we are going to discuss properties of lattices with many uniform elements. Further, we examine these properties in the case of lattices of submodules and of subgroups. We also formulate some questions related to the subject of this note.