Generalized Projective Modules And Injective Modules Over Formal Triangular Matrix Rings

Projective modules and injective modules play important roles in homo-logical algebras and representation theory of algebras. Recently, generalized projective modules and generalized injective modules were widely investigat-ed by many authors. In addition, formal triangular matrix rings as a natural generalization of rings are important. They are used to construct the counter examples. This makes the theory of rings and modules more abundant and concrete. This dissertation is inspired by combination of these knowledge, and it consists of three parts.First, we introduce the definition of modules over formal triangular ma-trix rings. The significance and the main works of this dissertation are also introduced.The second part introduced two ways to describe projective modules over formal triangular matrix rings. Submodules and quotient modules over trian-gular matrix rings are brought in as theoretical preparations. And, we further discuss the properties of morphisms between modules over triangular matrix rings. Consequently, three generalized projective modules over triangular ma-trix rings are discussed, say Gorenstein-projective modules, quasi-projective modules and pseudo-projective modules.In the last part, we will introduce our two main results. Different from separating injective modules into indecomposable, we given another way to describe injective modules over formal triangular matrix rings. Furthermore, Gorenstein-injective modules are explicitly described. Then, we gave some results about modules over formal triangular matrix rings to be quasi-injective and pseudo-injective.