Recall that a commutative ring $R$is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module $M$ is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.

Let R be a commutative ring with identity and $Spe{c}^s(M)$ denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by $\mathcal{O}(N,M)$, on $Spe{c}^s(M)$ equipped with the dual Zariski topology of M, where N is an R-module. We give a characterization of the sections of the sheaf $\mathcal{O}(N,M)$ in terms of the ideal transform module. We present some interrelations between algebraic properties of N and the sections of $\mathcal{O}(N,M)$. We obtain some morphisms of sheaves induced by ring and module homomorphisms.

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.