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The purpose of this paper is to introduce the concept of 2-absorbing coprimary submodule of a module as a generalization of strongly 2-absorbing second submodules and investigate some properties of this submodule class. We study the structure of 2-absorbing coprimary submodules of comultiplication modules, fully coidempotent modules and modules over a Noetherian ring. We also investigate some interrelations with 2-absorbing coprimary submodules, strongly 2-absorbing second submodules and strongly 2-absorbing secondary submodules.

Let R be a commutative ring with identity and $$\hbox {Spec}^{s}(M)$$ denote the set all second submodules of an R-module M. In this paper, we investigate various properties of $$\hbox {Spec}^{s}(M)$$ with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for $$\hbox {Spec}^{s}(M)$$ to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that \(M = \tau (M)\,\oplus\,K\) and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.