On the Weakly Second Spectrum of a Module

Abstract

In this paper, we extend the denition 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 dene the notion of weakly second radical of a submodule
and determine the weakly second radical of some modules. We also dene 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.