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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)
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