The conditions \((C_{i})\) in modular lattices, and applications (Q2788746)

The paper deals, within upper continuous modular lattices, with (pseudo-)complementation and the lattice-theoretic versions of the concepts of essential, closed, and uniform submodules. The first section gives an overview of (to a large extend known) results relating these concepts, Particular attention is given to the condition that any element has a pseudo-complement which is a direct summand. In Section 2, it is shown that this condition is inherited by direct sums and summands. Sections 3 and 4 transfer this into the settings of Grothendieck categories and module categories with hereditary torsion theory.