#TITLE_ALTERNATIVE# - (NIM 10104107), FAISAL
Modern algebra is introduced as the study of structures of mathematical system such as groups, rings and vector spaces. These structures come as the generalization of things we already know. For example, sets of real numbers R and complex numbers C inspired a mathematical system called field. Similarly, R2 and R3 vector spaces inspired the concept of generalized n-dimensional vector spaces. Beside that, there are another structures called algebras and modules. An algebra over field has two structures: the structure of vector space and of ring. Modules which we will discuss in here are modules over algebras. One of the results in the topics of algebras and modules over algebras is the Double Centralizer Theorem. If A is an algebra and and M an irreducible module over A then if AM is the representation of A in End(M) then the centralizer of AM's centralizer is AM itself. We will see the visualization of this theorem in some examples of algebras, i.e the group algebra of cyclic group of order 3 over C, C[C3] and the group algebra of dihedral group over C, C[D6]. The approach we use here is to translate the language of linear transformations into the language of matrix.