On the Weakly Second Spectrum of a Module

Abstract

In this paper, we extend the definition of weakly second submodule of a module over a commutative ring to a module over an arbitrary ring. First, we investigate some properties of weakly second submodules. We define the notion of weakly second radical of a submodule and determine the weakly second radical of some modules. We also define the notion of weak m*-system and characterize the weakly second radical of a submodule in terms of weak m*-systems. Then we introduce and study a topology on the set of all weakly second submodules of a module. We give some results concerning irreducible subsets, irreducible components and compactness of this topological space. Finally, we investigate this topological space from the point of view of spectral spaces