Characterization Of Derivations On Triangular Rings Or Prime Rings: All-Derivable Points

Let A be a ring (or an algebra) with unit I. Recall that an additive mapδfrom A into itself is called a derivation ifδ(AB)-δ(A)B+Aδ(B) for all A,B∈A;a mapδ:A→A is derivable at a given point Z∈A, ifδ(A)B+Aδ(B)=δ(Z) for any A, B∈A with AB=Z, and such Z is called an derivable point ofδ. An element Z∈A is an additive all-derivable point of a ring A if every additive map from A into itself that is derivable at Z is in fact a derivation. The purpose of this paper is to find some new additively all-derivable points for triangular rings and prime rings. As an application,additively all-derivable points for operator algebras, such as the nest algebras or factor von Neumann algebras are obtained. The followings are the main results.1.Let u=Tri(A,B,M) be a triangular ring, where A and B are unital rings with units I1and I2, respectively, and M is a faithful (A,B)-bimodule. Assume that, for every A∈A (B∈B), there exits some integer n such that nI1-A is invertible in A (nI2-B is invertible in B). Then, for any invertible element Z1∈A(Z2∈23), the element of the form is an additively all-derivable point of u. Assume that, for every A∈A and B∈B, there is some integer n such that nI1-A is invertible in A and nI2-B is invertible in B. Then, for any invertible element Z1∈A and Z2∈B,the element of the form is an all-derivable point of U.2.Let A be a ring with unit I and containing a non-trival idempotent P.Assume that,for every A∈A,there exits some integer n such that nI-A is invertible,and assume further that PAPA(I-P)=0 and PA(I-P)B(I-P)=0 implies PAP=0 and(I-P)B(I-P)=0.For everyΩ∈4,ifΩ=PQP is an invertible element in PAP(ifΩ=(I-P)Ω(I-P)is an invertible element in(I-P)A(I-P),thenΩis an all-derivable point of A.Note that the prime rings meet the condition of the above rerult.