Some Researches On Abelian Categories And"Trivial Extension"of Abelian Categories

In this thesis, by using the technique of categories, we study the endomorphism rings of indecomposable objects of finite length in k-abelian categories, ring-like trivial extensions of pre-additive categories and the trivial extensions of finitary categories as well as their Hall algebras respectively.At the beginning of this thesis, we first introduces the background of this paper, which includes the history and the trend of k-abelian categories, ring-like trivial extensions of pre-additive categories and the trivial extensions of finitary categories as well as their Hall algebras. Moreover, the framework of this dissertation will be presented.The first chapter is devoted to the Fitting lemma of abelian categories and a precise calculation formula for the endomorphism rings of indecomposable objects of finite length in k-abelian categories.In the second chapter, different from the work of R.Fossum, P. Griffith and I.Reiten in their study of trivial extensions of abelian categories, we originally intro-duce a new extension category for any given pre-additive categories, which is called the ring-like trivial extension of the pre-additive category. Based on this definition, some properties and results are given in this chapter.In the third chapter, we give a brief introduction of the trivial extensions of finitary categories and their Hall algebras.