Narrow Results

In this paper, we consider aspects of the big lattice of preradicals, related to pseudocomplements and supplements. We consider essential preradicals and superfluous preradicals, and we characterize the situation in which all nonzero preradicals are essential as well as the one in which all proper preradicals are superfluous. Also, we give two different proofs of the known fact that the big lattice of preradicals is strongly pseudocomplemented.

In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices $R$-tors (of hereditary torsion classes), $R$-nat (the lattice of natural classes) and $R$-conat (the lattice of conatural classes) coincide.

We characterize lattices and semilattices with the greatest element 1 where for every element p the interval [p, 1] is a pseudocomplemented lattice or a semilattice, respectively. If for elements x, y the relative pseudocomplement x * y exists then it coincides with the pseudocomplement of x in the interval [x ∧ y, 1]. However, the pseudocomplement of x in [x ∧ y, 1] can exist also when the relative pseudocomplement x * y does not. Thus our construction is a generalization of relative pseudocomplementation also to non-distributive lattices.