Notes on \((\alpha,\beta)\)-derivations (Q1380765)

An \((\alpha,\beta)\)-derivation of a ring \(R\), where \(\alpha\) and \(\beta\) are ring automorphisms of \(R\), is an additive map \(d\colon R\to R\) satisfying \(d(xy)=d(x)\beta(y)+\alpha(x)d(y)\) for all \(x,y\in R\). The paper studies \((\alpha,\beta)\)-derivations of (semi)prime rings satisfying some special relations. A sample result: Let \(R\) be a prime ring of characteristic not 2, \(d\) be a nonzero \((\alpha,\beta)\)-derivation of \(R\), \(U\) be a nonzero ideal of \(R\) and \(a,b\in R\). Then \(\alpha(a)d(x)=d(x)\beta(b)\) for all \(x\in U\) if and only if either \(\beta(b)=\alpha(a)\in C_R(d(U))\) or \(C_R(a)=C_R(b)=\{x\in R;\;d(x)=0\}\) and \(a[a,x]=[a,x]b\) (or \(a[b,x]=[b,x]b\)) for all \(x\in U\). Here, \(C_R(A)=\{x\in R;\;[a,x]=0\) for all \(a\in A\}\).