On topological lattices and their applications to module theory

Yassemi's "second submodules" are dualized and properties of its spectrum are studied. This is done by moving the ring theoretical setting to a lattice theoretical one and by introducing the notion of a (strongly) topological lattice L = (LΛV) with respect to a proper subset X of L. We investigate and characterize (strongly) topological lattices in general in order to apply it to modules over associative unital rings. Given a non-zero left R-module M, we introduce and investigate the spectrum Specf(M) of first submodules of M as a dual notion of Yassemi's second submodules. We topologize Specf(M) and investigate the algebraic properties of M by passing to the topological properties of the associated space.