The Osofsky-Smith Theorem in rings, modules, categories, torsion theories, and lattices

The renown Osofsky-Smith Theorem (O-ST), invented in 1991, says that a cyclic

(finitely generated) right R-module such that all of its cyclic (finitely generated) sub-
factors are CS modules is a finite direct sum of uniform submodules.

In this talk we present various extensions of this theorem to Grothendieck categories (the Categorical O-ST), module categories equipped with a hereditary torsion theory (the Relative O-ST), and modular lattices (the Latticial O-ST); it illustrates a general strategy which consists on putting a module-theoretical concept/result into a latticial frame (we call it latticization) in order to translate that concept/result to Grothendieck categories (we call it absolutization) and to module categories equipped with a hereditary torsion theorie (we call it relativization).