The Osofsky-Smith Theorem for Modular Lattices, and Applications (II)

Strongly Extending Modular Lattices

“…This section is allocated to employ the main results in Section 3 to Grothendieck categories. First, we recall some notations and terminology from [1][2][3][4][5][6][7][8][9][10][11]. In this section G will indicate a Grothendieck category.…”

“…Let H be an object of G. We will denote by L(H), the upper continuous modular lattice of all subobjects of H ( [11], [21, Chapter 4, Proposition 5.3, and Chapter 5, Section 1]). According to [2], for any object H of G, and for each subset W ⊆ L(H), we denote…”

“…In Section 2, we recall some preliminaries and definitions about lattices from [1][2][3][4][5][6][7][8][9][10][11]. We recall the useful notion of linear morphisms between two lattices introduced by Albu and Iosif [5].…”

On the De Morgan’s Laws for Modules

“…Like in the case of topological spaces, the behavior of a module can be done via its lattice of submodules as it has been explored in [Alb14b,AIT04,Alb14a,Sim14b,CRT19b].…”

Attaching topological spaces to a module (I): Sobriety and spatiality

“…For the study of a module categories over a unitary ring one technique is through the examination of complete lattices of submodules, such as in [Alb14b], [Alb14a], [Simc] and [MSZ16]. Following this idea, we give a module-theoretical counterpart of the ring-theoretical examination developed by H. Simmons in [Sim89] and [Sima].…”