On topological lattices and their applications to module theory

Abstract

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 [Formula: see text] with respect to a proper subset [Formula: see text] of [Formula: see text] 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 [Formula: see text]-module [Formula: see text] we introduce and investigate the spectrum [Formula: see text] of first submodules of [Formula: see text] as a dual notion of Yassemi’s second submodules. We topologize [Formula: see text] and investigate the algebraic properties of [Formula: see text] by passing to the topological properties of the associated space.