A note on commutativity of semiprime Banach algebras

In the present paper it is shown that if R is a semiprime ring of characteristic different from two which admits a derivation d such that \([d(x^m),d(y^n)]=\pm [x^m, y^n]\) for all \(x,y\in R\), where \(m\ge 1, n\ge 1\) are fixed positive integers, then R is commutative. Further using this result it is established that if \(\mathfrak {A}\) is a semiprime Banach algebra and \(\mathscr {H}_1\) and \(\mathscr {H}_2\) are nonvoid open subsets of \(\mathfrak {A}\) which admits a continuous derivation \(d:\mathfrak {A}\rightarrow \mathfrak {A}\) such that \([d(x^m),d(y^n)]\pm [x^m,y^n]=0\) for all \(x\in \mathscr {H}_1\) and \(y\in \mathscr {H}_2\), where m, n are no longer fixed rather they depend on the pair of elements \( x, y \in \mathfrak {A}\), then \(\mathfrak {A}\) is commutative.