Trigonometric Algebra And The (α, β)-derivable Mapping On The Ring

In this thesis,we study the characterization of(α,β)-derivable mappings on triangular algebras and rings with idempotents.The thesis is organized follows:In Chapter 1,we fix some notation and recall necessary definitions needed in the thesis(for example,triangular algebras,derivations,(α,β)-derivations and so on).In Chapter 2,we mainly study(α,β-derivable mappings on t riangular al-gebras by Jordan product zero(idempotent or unital)elements.Specifically,let u=Tri(A,M,B)be a triangular algebra,and α,β u→u be automorphisms.In this paper,we prove that if δ:u→u is a linear map satisfyingδ(XY)=δ(X)β(Y)+α(X)δ(Y)for any X,Y ∈u with X ο Y=0(resp.X ο Y=P1 or X ο Y=11),then δ(X)=T(X)+α(X)δ(I)for any X∈u,where T:u→u is a(α,β)-derivation,δ(I)∈ Zα,β(u)(resp.δ is a(α,β)-derivation).In Chapter 3,we mainly discuss(α/β)-derivable mappings on rings with non-trivial idempotents,and obtain(α,｛)-derivable mappings are(α,β)-derivation.Specif-ically,let,A be a unital ring with unit I and nontrivial idempotent P1.Assume that,for every A ∈A A,there exists some integer n such that nI-A is invertible,and assume further that A ∈ A such that(1)if α(P1)Aβ(A12)={0},then α(P1)Aβ{P1)=0;(ii)if a(A12)Aβ(P2)={0},then α(P2)Aβ(P2)=0,where α,β:A →A be automorphisms.In this paper,we prove that,if δ:A→A is a linear map satisfying δ(AB)=δ(A)β(B)+α(A)δ(13)for any A,B ∈ A with AB=P1,then δ is a(α,β)-derivation.