Research On The Structure Of Generalized Jordan*-Derivations On Rings

A linear anti-automorphism of period 2 on an algebra A is called an involution,which is generally represented by*,and an algebra equipped with an involution is called a*algebra.Similarly,a ring containing an involution is called a*ring.A linear mapping D over the*algebra A that satisfies D(x2)=D(x)x*+xD(x)for any x?A is called a Jordan*-derivative over A.In the study of Jordan*-derivatives on*algebras,scholars had found that the structure of Jordan*-derivatives is closely related to whether the doubly semi-linear functional can be represented by quadratic functional.This paper first introduces the background and significance of the topic,and gives the research on algebra and various derivations on the ring by scholars at home and abroad;then introduces the basic concepts used in this article,and explains it with examples;finally obtains:generalized Jordan*-derivation D is non-trivial on the ring Rn,D is trivial on the ring Sn,the generalized Jordan*-derivations D satisfying the conditions D(x)=-xD(x-1)x*are trivial in rings Mn(C).