Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings.

*(English)*Zbl 0791.16028Using 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 \(\gamma([R,R]) =0\).

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 \(\gamma([R,R]) =0\).

Reviewer: C.Lanski (Los Angeles)

##### 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) |