Multiplicative Lie Derivationsand Multiplicative Jordan Derivations Of Triangular 3-matrix Rings

Derivations,Jordan derivations,Lie derivations and relationship between them are very important research topics in operator algebra and operator theory.An additive map φ on an associative ringRis called a derivation if φ(AB)= φ(A)B + Aφ(B)holds for all A,B∈ R;is called a Jordan derivation if φ(AB + BA)= φ(A)B + Aφ(B)+ φ(B)A + Bφ(A)holds for all A,B∈ R;is called a Lie derivation if φ(AB-BA)= φ(A)B-Bφ(A)+ Aφ(B)-φ(B)A holds for all A,B∈ R.If the additive assumption on φ is deleted,the above map is called respectively a multiplicative derivation,a multiplicative Jordan derivation and a multiplicative Lie derivation.In this paper,we discuss the structural properties of multiplicative Lie derivations and multiplicative Jordan derivations and their relations to the derivations in the pure algebraic framework of triangular 3-matrix rings.We proved that :(1)with the standard hypothesis on the center,every multiplicative Lie derivation on a triangular 3-matrix ring has the standard form;that is,the sum of a derivation and a center-valued map vanishing on each commutator;(2)every multiplicative Jordan derivation on a 2-torsion free triangular 3-matrix ring is a derivation;(3)every multiplicative Jordan derivation on a 2-torsion triangular ring or on a 2-torsion triangular 3-matrix ring is a sum of a derivation and a center-valued map vanishing on each Jordan product.