Hypercommuting conditions of b-generalized skew derivations on Lie ideals in prime rings

Let R be any non-commutative prime ring of char \((R)\ne 2\), L a non-central Lie ideal of R and F, G be b-generalized skew derivations of R. Suppose that

$$[F(u)u-uG(u), u]_n=0$$

for all \(u\in L\) and for some fixed integer \(n\ge 1\), then one of the following assertions holds:

1. (1)

   there exist \(a'',b''\in Q_r\) such that \(F(x)=xa''\), \(G(x)=b''x\) for all \(x\in R\) with \(a''-b''\in C\);

2. (2)

   \(R\subseteq M_2(K),\) the algebra of \(2\times 2\) matrices over a field K and

   • either K is a finite field;

   • or there exists \(\lambda \in C\) such that \((F+G)(x)=\lambda x\) for all \(x\in R\);

   • or there exists \(\lambda \in C\) and \(h\in Q_{r}\) such that \((F+G)(x)=hx+xh+\lambda x\) for all \(x\in R\).

The above result, naturally improves the recent result obtained by Carini et al. in [4].