A topology and the frame attached to a set of primitive submodules

Ponente(s): Jaime Castro Pérez, José Ríos Montes

For a multiplication R-module M we define the primitive topology T on the set Prt (M) of primitive submodules of M. We prove that if R is a commutative ring and M is a multiplication R-module, then the complete lattice Sprt (M) of semiprimitive submodules of M is a spatial frame. When M is projective in the category σ[M], we obtain that the topological spaces (Prt(M), T ) and (Prt(R), T ) are homeomorphic. As an application, we prove that if M is projective in the category σ[M], then Prt(R) has classical Krull dimension if and only if Prt(M) has classical Krull dimension.