Module Categories And Homological Properties Of A Class Of Matrix Algebras

Triangular matrix algebras are a very important kind of algebras including one-point extension algebras,which have attracted the attention and in-depth research of many experts and scholars.In this dissertation,we generalize the concept of triangular matrix algebras,and introduce matrix algebras A=(?)and two-point extension algebras B[M,N]=(?).Then we study their module categories,homological properties and Gorenstein relative homological properties.The main content of this dissertation is divided into the following parts:In the first chapter,we introduce the research background and the main works of this dissertation,as well as the description of some symbols.In the second chapter,we mainly study the module category of ? and its homological properties.Firstly,we give the definition and examples of the matrix algebras.Secondly,we characterize the module category of ?,and prove that the category A-mod is equivalent to each of the categories ?C and ?C.Finally,we characterize the simple module,the indecomposable projective module,the indecomposable injective module and global dimension of A.In the third chapter,we mainly study the Gorenstein relative homological properties of ?,and describe the structure of Gorenstein injective module.In the fourth chapter,we mainly study two-point extension algebras.We give a necessary and sufficient condition for a finite-dimensional k-algebra to be a two-point extension algebra,then we describe the global dimension of B[M,N].Finally,we prove that if two algebras are Morita equivalent,their corresponding two-point extension algebras are also Morita equivalent.