Lattice preradicals with applications to Grothendieck categories and torsion theories.
The authors define a lattice preradical. This is based on the notion of a linear morphism of lattices introduced earlier by them. As the name suggests, such morphisms for lattices capture the fundamental properties of a linear mapping between modules. At first a preradical is defined on the category \(\mathcal{LM}\) of linear modular lattices. The main result here is that this notion is compatible with direct joins in the sense that a preradical commutes with arbitrary direct joins. The definition of a preradical is then slightly generalized to hereditary full subcategories of \(\mathcal{LM}\). This generalization is used to give lattice theoretic counterparts of the well-known module preradicals trace and reject. The other well-known radicals like the socle and the Jacobson radical are also introduced in this general setting and the relationships between them are explored. Finally it is shown that this approach to preradicals for lattices lends itself to a smooth and easy interpretation in Grothendieck categories as well as module categories with a hereditary torsion theory.
scientific article; zbMATH DE number 6493632
scientific article