Two results concerning symmetric bi-derivations on prime rings
Let \(R\) be a prime ring with \(\text{char}(R)\neq 2,3\), and \(D\colon R\times R\to R\) a bi-additive, symmetric map so that \(D(\;,y)\colon R\to R\) is a derivation for all \(y\in R\). The paper proves two results about such mappings, called symmetric bi-derivations. The first assumes that \(D_ 1\) and \(D_ 2\) are symmetric bi-derivations with \(D_ 1(x,x)D_ 2(x,x)=0\) for all \(x\in R\) and proves that \(D_ 1=0\) or \(D_ 2=0\). The second result shows that if \(D\) is a symmetric bi-derivation with \([D(x,x),x]\) central for all \(x\in R\), then either \(R\) is commutative or \(D=0\).
scientific article; zbMATH DE number 4179581
scientific article