Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called special selfgenerator or weak multiplication module if for each cyclic submodule Ra of M (equivalently, for each submodule N of M) there exists a family {fi} of endomorphism of M such that Ra = ∑_i▒f_i (M) (equivalently N = ∑_i▒f_i (M)). In this paper we introduce a class of modules properly contained in selfgenerator modules called special selfgenerator modules, and we study some of properties of these modules.

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

In this paper, we apply the notion of a bipolar fuzzy n-fold KU-ideal of KU- algebras. We introduce the concept of a bipolar fuzzy n-fold KU-ideal and investigate several properties. Also, we give relations between a bipolar fuzzy n- fold KU-ideal and n-fold KU-ideal. The image and the pre-image of bipolar fuzzy n-fold KU-ideals in KU-algebras are defined and how the image and the pre- image of bipolar fuzzy n-fold KU-ideals in KU-algebras become bipolar fuzzy n- fold KU-ideals are studied. Moreover, the product of bipolar fuzzy n-fold KU- ideals in Cartesian product KU-algebras is given.

This paper refers to studying some types of ideals, specifically cubic bipolar ideals and cubic bipolar T-ideals of TM algebra. It also introduces a cubic bipolar sub-TM-algebra and several important properties of these concepts. The relationships between these ideals and characterizations of cubic bipolar T-ideals are investigated.

In This paper, we introduce the associated graphs of commutative KU-algebra. Firstly, we define the KU-graph which is determined by all the elements of commutative KU-algebra as vertices. Secondly, the graph of equivalence classes of commutative KU-algebra is studied and several examples are presented. Also, by using the definition of graph folding, we prove that the graph of equivalence classes and the graph folding of commutative KU-algebra are the same, where the graph is complete bipartite graph.

This study investigated the cubic intuitionistic fuzzy set of TM-algebra as a generalization of the cubic set. First, a cubic intuitionistic ideal and a cubic intuitionistic T-ideal are defined, followed by a discussion of their properties. Furthermore, the level set of a cubic intuitionistic TM-algebra is defined, and the relationship between a cubic intuitionistic level set and the cubic intuitionistic T-ideal is established. A novel definition of a cubic intuitionistic set under homomorphism is proposed, and several significant results are demonstrated.