Some Studies On Path Algebras

Let KQ denote the path algebra of a quiver Q over a field K. In this thesis we study the relationship between the geometric properties of Q and the algebraic properties of KQ. In terms of the geometric properties of Q, we give some necessary and sufficient conditions for KQ being a finite-dimensional algebra, a primitive algebra, a right Noether algebra, respectively, and give a concrete description of the radical rad KQ and a K-linear representation of Q. Finally we give the standard decomposition theorem of KQ—modules and its applications.