Left Derivations And Jordan Left Derivations On Rings

The study of derivations,which are important both in theory and applications,is an active topic in operator algebra and operator theory.In this thesis,we maily discuss left derivations and Jordan left derivations on triangular rings and prime rings,and give some characterization of left derivations and Jordan left derivations from different aspects.The following are the main results obtained in this thesis.1.Let R be a prime ring containing identity I and a non-trivial idempotent element E.Suppose that the characteristic of R is neither 2 nor 3,1/2I ∈ R,and for any A ∈R,there exists a positive integer n such that nI-A is invertible in R.Assume that δ:R→L is an additive map satisfying Aδ(B)+ Bδ(A)=δ(AB)whenever AB = I(AB = 0)for any A,B ∈ R.Then d must be zero.2.Let A and B be rings with characteristic not 3,and IA,1/2IA ∈ A,IB,1/2IB ∈B.Let Mbe an(A,B)-bimodule,which is faithful as a right B-module,and let U = Tri(A,M,B)be the triangular ring.Suppose that δ:u→u is an additive map and G ∈ u is any fixed element.(i)If G = 0,then δ satisfies δ(XY)= Xδ(Y)+Yδ(X)whenever XY = 0 for any X,Y ∈ u if and only if there exist additive maps δ1:A→A and δ2:A→ M,which satisfy δi(A1A2)= A1δi(A2)+ A2δi(A1)(i = 1,2)when A1A2 = O for any A1,A2 ∈ A,such that .(ii)If G ≠ 0,and if for any A ∈ A and B ∈ B,there exist some positive integers n,m such that nlA-A and mIB-B are invertible in A and B,respectively,then δsatisfies δ(XY)= Xδ(Y)+Yδ(X)whenever XY = G for any X,Y ∈ u if and only if:(a)Qδ(PXP)= 0 holds for all X ∈ u and δ(PGP)= PXPδ(PYP)+ PYPδ(PXP)holds for all X,Y ∈ U with XY = G;(b)δ(PGQ)= δ{QGQ)= 0,and δ(PXQ)= PXQδ(Q)and δ(QXQ)= QXQδ(Q)hold for all X ∈ u;(c)YXδ(Q)0 holds for all X,Y ∈ u with XY = G.3.Let A and B be two unital rings,and let M be an(A,B)-bimodule which is faithful as a right B-module.Let u = Tri(A,M,B)be the triangular ring.Assume thatδ:u→u is a map.Then δ is a multiplicative left derivation if and only if there exist two multiplicative left derivations δ1:A→A and δ2 A → M such that for al ;δis an additive Jordan left derivation if and only if there exist additive Jordan left derivations δ11:A→A,δ12:A →M,δ22：A→B,τ11:B→A,τ12:B→M,τ22:B→B,which satisfy 2τ11(B)=2τ12(B)= 2τ22(B)= 2δ22(A)= 0 for all A ∈ and B ∈ B,suuh that for all4.Let R be a unital prime ring with a non-trivial idempotent.Then there does not exist nonzero multiplicative left derivations on R;in addition,if the characeristic of R is not 2,then there does not exist nonzero additive Jordan left derivations on R.