| Let A be an associative algebra. Studying the (non-) linear mappings of algebraA is helpful for understanding its structural information and classification. Morespecifically, we usually concern with two kinds of mappings: one kind is the mappingpreserving algebraic operations, such as isomorphisms, Lie isomorphisms, Jordanisomorphisms etc.; the other kind contains various mappings that satisfy the Leibnizformula, such as derivations, Lie derivations, Jordan derivations etc. The principalpurpose of this dissertation is to study the following question: under what conditionsor on which algebras, Jordan higher derivations will degenerate to be higher deriva-tions?Firstly, we study Jordan higher derivations from the point of view of the innerhigher derivations. We will recall the conclusion that each Jordan higher derivation ofa torsion free ring can be written as a linear combination of compositions of Jordanderivations. Then we will introduce two kinds of equivalent definitions of innerhigher derivations. Finally, we proved that if every Jordan derivation of thealgebra A is inner, then every Jordan higher derivation of A is also inner.Secondly we will consider the aforementioned problem from the perspective ofJordan homomorphisms. We will generalize one main result of Jacobson and Rickart[1]about the relationship between Jordan derivations and Jordan homomorphisms. Inparticular, we will describe explicitly the form of Jordan higher derivations on the fullmatrix algebra over a noncommutativi ring.Last, we will consider the aforementioned problem from the perspective ofHochschild2-cocycles. We first briefly introduce the Hochschild cohomology. Usingthe skew-symmetric Hochschild2-cocycles we proved that: if the associativesubalgebra of A generated by the second derived Lie algebra is equal to A itself, theneach Jordan higher derivation will degenerate to be a higher derivation. Finally, wecan apply this method to semi-prime rings, and give a new proof of the main theoremof [2]. |
