Artin cebirlerinin temsil teorisindeki bazı kategori teori metotları

During the last decades there has been a significant development in the representation theory of Artin algebras due to different techniques which have been proved to be fruitful while dealing with problems of that nature. The main aim of this Master Thesis is to discuss some of them and see how they can be applied to a particular open problem of our own choice. Such a problem is the conjecture established in the end of the book "Representation theory of Artin algebras" by Auslander, Reiten and Smalo which asserts that an Artin algebra with the property that its finitely generated indecomposable modules are completely determined by its composition factors is of finite representation type. Examples of rings with this property are the semisimple artinian rings and the rings of the form $\mathbb{Z}_n$. We will be able to prove the conjecture for some special cases which include the commutative, the hereditary and the $J^2=0$ case. Indeed, in the first two cases we will show that the converse also holds. We will also give a generic example of an Artin algebra with $J^2=0$ and of finite representation type but whose finitely generated indecomposable modules are not completely determined by their composition factors.