Zariski topology for multiplication modules with applications to frames, quantales and classical Krull dimension

Abstract

For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M=N) of semiprime submodules of M=N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval [N;M] = fP 2 Semp (M) j N Pg and the lattice Semp (M=N) are isomorphic as frames. Finally, as applications we obtain results about quantales and the classical Krull dimension of M.