About Some Of The Categories Of Modules And Homological Characteristics Of Triangular Matrix Rings

This paper consists of three chapters. The aim of this paper is to study the category of modules over a triangular matrix rings and its homological properties. In the first chapter, we introduce some concepts, the background of this issue and some intersting known results. Then we tell readers what we will do in this paper. In the second chapter, firstly, the definition of triangular matrix rings Γ, which is of order 3, is introduced. By establishing an equivalent functor , it is shown the equivalence between the category of finitely generated modules mod Γ and the category Γ^L (see the definition 2.5). Then, we determine the simple objects and indecomposable projective objects in Γ^L completely. As a byproduct, we obtain a estimate of the global dimension of Γ. Morever, we give the Jacobason radical of Γ and an application of the triangular matrix algebra. In the third chapter, firstly , we introduce the concept of triangular matrix ring of order n, which is not a direct generalization of triangular matrix rings of order 3. We denote it by Γ_n. Then, the equivalence between mod Γ_n, the category of finitely generated Γ_n-modules, and the category F_n^L(see the definition 3.1), is also obtained. Similar to the case of order 3, the Jacobason radical of Γ_n, the simple objects and indecomposable projective objects of Γ_n^L are determined,we also give a estimate of the global dimension of Γ_n. In the fourth chapter, we characterize respectively uniform , hollow moduies over Γ₃, also obtain Rad(A(?)B(?)C) and Soc(A(?)B(?)C).