One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

The main purpose of this article is to classify all indecomposable quasi-semiprime comultiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi-semiprime comultiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi-semiprime comultiplication modules and classify quasi-semiprime comultiplication modules over local Dedekind domains. Second, we get all indecomposable Separated quasi-semiprime comultiplication modules and then, using this list of Separated quasi-semiprime comultiplication modules, we show that non-Separated indecomposable quasi-semiprime comultiplication Rmodules with finite-dimensional top are factor modules of finite direct sums of Separated indecomposable quasi-semiprime comultiplication R-modules.

Let R be a commutative ring and M be a unitary R- module. We define a semiprime submodule of a module and consider various properties of it. Also we define semi-radical of a submodule of a module and give a number of its properties. We define modules which satisfy the semi-radical formula (s.t .s.r.f) and present the existence of such a module.

The monoid Cb of name substitutions and the notion of finitely supported Cb-sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported Cb-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported Cb-set is an injective map. Also, for every two distinct elements of a simple finitely supported Cb-set, there exists an element of the monoid Cb which separates them by making just one of them into an element with the empty support. In this paper, we generalize these properties of simple finitely supported Cb-sets by modifying slightly the notion of the support map,defining the notion of 2-equivariant support map,and introducing the notions of s-Separated and z-Separated finitely supported Cb-sets. We show that the notions of sSeparated and z-Separated coincide for a finitely supported Cb-set whose support map is 2-equivariant. Among other results, we find a characterization of simple s-Separated (or z-Separated) finitely supported Cb-sets. Finally, we show that some subcategories of finitely supported Cb-sets with injective equivariant maps which constructed applying the defined notions are reflective.

In this paper, by considering the notion of prime BCK- sub-modules of BCK-modules, we introduce a topology on prime BCK-sub-modules of BCK-modules. Moreover, the notion of top BCK-modules, semi-prime sub -BCK-modules and extraordinary sub-BCK-modules are introduced. Finally the relationships between them are studied.