Characterizing Derivations On Triangular Rings By Local Behaviors

In the present paper, we study the problem what kind of local behavior of additive maps on a trianglar ring will ensure they are additive derivations. Let<u=Tri(A,M,B) be a triangular ring and G ∈U. Recall that X o Y= XY+YX is the Jordan product of X, Y ∈ U. Let Φ:uâ†'u be an additive map. We say that Φ is Jordan derivable at G if Φ(X o Y)=Φ(X) o F+X oΦ(Y) for all X,Y ∈ u with X o Y= G. G is called a Jordan all-derivable point of u if every additive map Jordan derivable at G is an additive Jordan derivation. We say that Φ is quasi Jordan derivable at G if Φ(XoY)=Φ(X)oY+XoΦ(Y) for all X, Y ∈u with XY= G.G is called a quasi Jordan all-derivable point of u if every additive map quasi Jordan derivable at G is an additive Jordan derivation. Under some assumptions on triangular ring u=Tri(A,M,B), it is shown that all elements G ∈ u are quasi Jordan all-derivable points of ty; every element G ∈u of the form ((A0) (O0)) or ((O0)(OB)) with A and B respectively in F(A) and (?)(B) is an additive Jordan all-derivable point of u, where (?)(A) denotes the center of A.