On the Primary Spectrum of Lattice Modules

Abstract

Consider a lattice module M over a C-lattice L. A proper element P of a lattice module M is called primary if aAleq P implies Aleq P or aⁿ1Mleq P for some positive integer n, for all a∈ L and A∈ M. In this work, we construct a topological framework on Specᵖʳ(M), the family of all primary elements of M. We examine several topological properties of this space, particularly compactness and irreducibility, and establish criteria under which Specᵖʳ(M) satisfies the conditions of a spectral space.