# zbMATH — the first resource for mathematics

Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. (English) Zbl 0791.16028
Using intricate but elementary calculations, the author obtains characterizations of Lie isomorphisms, Lie derivations, and other mappings of prime rings which extend a number of results in the literature. For the statement of all the results which follow, $$R$$ and $$A$$ are prime rings with neither of characteristic two and neither embedding in $$M_ 2(K)$$ for $$K$$ a field. The extended centroid of $$R$$ is denoted by $$C(R)=C$$, and $$[x,y]=xy-yx$$.
The main results are: Theorem 1. If $$f: R\times R\to R$$ is biadditive and $$[f(x,x),x]=0$$ for all $$x\in R$$, then $$f(x,x) =\lambda x^ 2+ \mu(x)x+ \nu(x)$$ for $$\lambda\in C$$ and $$\mu,\nu:R\to C$$ with $$\mu$$ additive; Theorem 2. Let $$R$$ and $$A$$ be centrally closed algebras over the field $$F\neq GF(3)$$, and $$\theta: R\to A$$ a bijective $$F$$-linear map satisfying $$[\theta(x^ 2), \theta(x)]=0$$ for all $$x\in R$$. Then $$\theta(x)= cg(x)+ h(x)$$ with $$c\in F-\{0\}$$, $$g,h: R\to A$$, $$g$$ is an isomorphism or anti-isomorphism onto $$A$$, and $$h(R)$$ is central; Theorem 3. If $$\theta: R\to A$$ is a Lie isomorphism ($$\theta([x,y])= [\theta(x),\theta(y)]$$), then $$\theta= \varphi+\tau$$ where $$\varphi: R\to AC(A)$$, $$\varphi$$ is a monomorphism or the negative of an anti-monomorphism, and $$\tau: R\to C(A)$$ with $$\tau([R,R])=0$$; and last, Theorem 4. If $$D$$ is a Lie derivation of $$R$$ ($$D([x,y])= [D(x),y]+ [x,D(y)]$$), then $$D= \delta+\gamma$$ where $$\delta: R\to RC(R)$$ is a derivation, and $$\gamma: R\to C$$ is additive with $$\gamma([R,R]) =0$$.

##### MSC:
 16W20 Automorphisms and endomorphisms 16N60 Prime and semiprime associative rings 16W25 Derivations, actions of Lie algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16U70 Center, normalizer (invariant elements) (associative rings and algebras)
Full Text: