Mutation Groups, P-derivations And Preservability Of Duality Of Quiver

In this thesis, we investigate three aspects questions as following:In Chapter 3, we generalize ordinary derivation and Jordan derivation of a ring to the concepts of P-derivation and P-Jordan derivation of a ring respectively. Some properties, e.g. the composition of P-derivations and P-Jordan derivations, are giv-en. Moreover, it proved that if R/P(R) is 2-torsion free for a ring R and its prime radicalP(R), P-Jordan derivations of R are P-derivations. More mainly, some con-clusions on the commutativity of R/I and R/P(R) are obtained via P-derivations and P-Jordan derivations, for a prime ideal I of R.In Chapter 4, we give sufficient conditions and necessary conditions on du-ality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. Meantime, we characterize the condition of generalized path algebras as cleft extensions of path algebras.In Chapter 5, we give the definition of mutation group for a set of skew-symmetric-matrices. The coxeter-indice of the mutation groups are calculated, meantime, some properties of mutation groups are provided.