On generalized \((\sigma,\tau)\)-biderivations in rings. (Q2891017)

The authors prove a number of special and technical results concerning generalized versions of derivations and Lie ideals. Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\) and center \(Z(R)\). If \(\sigma,\tau\in\Aut(R)\), let \(U\) be a nonzero \((\sigma,\tau)\)-Lie ideal of \(R\), \(V\) a noncentral square closed Lie ideal of \(R\), \(B\colon R\times R\to R\) a biadditive and symmetric mapping, and \(f\colon R\to R\) given by \(f(x)=B(x,x)\). The first results show that \(f(U)=\{0\}\) or \(U\subseteq Z(R)\) under certain assumptions. Two of several of these assumptions are that there are \(\alpha\in\Aut(R)\) and \(\beta\in\text{End}(R)\) so that for all \(u,v\in U\), either \(f(uv)=f(u)\alpha(v)+\beta(u)f(v)\) or \(f(uv)=\alpha(v)f(u)+\beta(u)f(v)\). Variations of these conditions also force the same conclusions. Similar results hold when \(U\) is replaced by a nonzero left ideal of \(R\).NEWLINENEWLINE Call \(B\) a \((\sigma,\tau)\)-biderivation if \(B(xy,z)=B(x,z)\sigma(y)+\tau(x)B(y,z)\) for all \(x,y,z\in R\) and then call a symmetric biadditive \(G\) on \(R\) a symmetric generalized \((\sigma,\tau)\)-biderivation when \(G(xy,z)=G(x,z)\sigma(y)+\tau(x)B(y,z)\). For such maps, the authors give conditions that imply \(G(xy,z)=G(x,z)\sigma(y)\), namely if for any \(x\in V\) either \([G(x,x),\sigma(x)]=0\), or if for any \(x,y\in V\), \(G(x,y)=[\sigma(x),\sigma(y)]\). For the latter condition, \(V\) may be replaced by a nonzero left ideal of \(R\). The final result in the paper shows that a symmetric, generalized, Jordan \((\sigma,\tau)\)-biderivation on \(V\) is a symmetric generalized \((\sigma,\tau)\)-biderivation on \(V\).