The Osofsky-Smith theorem [B. L. Osofsky and P. F. Smith, J. Algebra 139, No. 2, 342–354 (1991; Zbl 0737.16001)] asserts that a cyclic (finitely generated) right \(R\)-module such that all of its cyclic (finitely generated) subfactors are extending modules is a finite direct sum of uniform submodules. The paper under review is a survey of various extensions of this theorem to Grothendieck categories, module categories equipped with a hereditary torsion theory, and modular lattices that appeared previously in papers devoted to this subject. These are [S. Crivei et al., Glasg. Math. J. 52A, 61–67 (2010; Zbl 1222.16005); the author, Commun. Algebra 39, No. 12, 4488–4506 (2011; Zbl 1262.06003); ibid. 42, No. 6, 2663–2683 (2014; Zbl 1285.06003)], cited in an incomplete chronology presented in the paper.
As early as 2000, the reviewer stated in his book [Lattice concepts of module theory. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0959.06001)] a lattice version of the Osofsky-Smith theorem (see Chapter 8, Theorem 8.4) and “paved the way” for a proof. Many lattice concepts presented in the paper already appeared in this book (which does not seem to be known to the author of this paper), including the so-called H-Noetherian lattices (i.e., compact elements form an ideal) and the so-called restricted socle condition (i.e., for each \(1\neq b\) essential, the quotient sublattice \(1/b\) has atoms).