On generalized biderivations and related maps
Using elementary but ingenious calculations, the author characterizes some interesting special types of mappings in prime rings. Let \(R\) be a prime ring, \(I\) an ideal of \(R\), \(Z\) the center of \(R\), \(C\) the extended centroid of \(R\), and \(Q\) the symmetric Martindale quotient ring of \(R\). If \(\sigma\in\text{Aut }R\), then an additive map \(D:R\to R\) is a \(\sigma\)-derivation if for all \(x,y\in R\), \(D(xy)=\sigma(x)D(y)+D(x)y\), and is a derivation when \(\sigma=\text{id}_R\). A map \(F:R\times R\to R\) is a \(\sigma\)-biderivation if for each \(x\in R\), \(F(x,\;)\) and \(F(\;,x)\) are \(\sigma\)-derivations. Call an additive map \(H:I\to R\) a generalized derivation if \(H(x)=ax+xb\) for some \(a,b\in R\) and all \(x\in I\), and \(E:I\times I\to R\) a generalized biderivation if for all \(x\in I\), \(E(x,\;)\) and \(E(\;,x)\) are generalized derivations. Finally let \([x,y]=xy-yx\). The four main theorems of the paper are as follows. Theorem A. If \(F\) is a \(\sigma\)-biderivation, then either \(D=0\), \(R\) is commutative, or for some \(q\in Q\) and all \(x,y\in R\), \(\sigma(x)=qxq^{-1}\) and \(F(x,y)=q[x,y]\). Theorem B. If \(E:I\times I\to R\) is a generalized biderivation, then there are \(a,b\in Q\) so that for all \(x,y\in I\), \(E(x,y)=xay+ybx\). Theorem C. Let \(f_1,\dots,f_4:I\to R\) be additive, and define \(F:I\times I\to R\) by \(F(x,y)=f_1(x)y+xf_2(y)+f_3(y)x+yf_4(x)\). If \(F(x,y)\in Z\) for all \(x,y\in I\), and if \(\text{char } R\neq 2,3\), then either \(R\) embeds in \(M_2(K)\) where \(K\) is a field, or \(F(x,y)=0\). Whenever \(F(x,y)=0\), for all \(x,y\in I\), there are \(a,b\in Q\) and additive \(z,t:I\to C\) so that \(f_1(x)=-xa+z(x)\), \(f_2(x)=ax-t(x)\), \(f_3(x)=-xb+t(x)\), and \(f_4(x)=bx-z(x)\). Theorem D. Let \(T\) be a right ideal of \(R\) and \(f:T\to R\), additive. Unless \(TC\) is a minimal right ideal in \(RC\) so that \(eTCe=eC\) when \(e^2=e\) and \(eRC=TC\), the following hold: 1) If \(D:T\times T\to R\) is a biderivation, then for some \(c\in C\) and all \(x,y\in T\), \(D(x,y)=c[x,y]\); 2) If \([f(x),x]=0\) for all \(x\in T\), then \(f(x)=cx+z(x)\) for \(c\in C\), \(z:T\to C\) additive, and all \(x\in T\); 3) If \(xf(x)=0\) for all \(x\in T\), then \(f=0\); and 4) If \(\text{char }R\neq 2\) and \(f(x)x+xf(x)=0\) for all \(x\in T\), then \(f=0\). Furthermore, 2), 3), and 4) hold for any \(T\) when \(f:T\to T\). Although computational, the paper is clearly written and easy to follow.
scientific article; zbMATH DE number 763927