Power central values with generalized derivations on Lie ideals of prime rings

Abstract

Throughout the work, \(\Re \) is a prime ring which is non-commutative in structure with characteristic different from two, where the center of \(\Re \) is \({\mathcal {Z}}(\Re )\). The rings \(Q_r\) and \({\mathcal {C}}\) are Utumi ring of quotients and extended centroid of \(\Re \) respectively. Consider \({\mathcal {P}}\) to be a Lie ideal of \(\Re \) which is non-central. Assume, the generalized derivation defined on \(\Re \) be \({\mathcal {K}}\) with associated derivation \(\mu \). If \({\mathcal {K}}\) satisfies certain typical power central functional identities along with an annihilator, then we have established the following: For instance, \(0 \ne e \in \Re \) with \(e({\mathcal {K}}(t)t)^m \in {\mathcal {C}}\) for every \(~t \in {\mathcal {P}} \) and \(m>0\) a fixed integer. Then one of the following conditions hold:

(i):

\({\mathcal {K}}(t)=qt\), \(q=a+b\) with \(a, b \in Q_r\), \(b \in {\mathcal {C}}\) and \(e=\beta ea\), where \(\beta =-b^ {-1}\), provided \({\mathcal {K}}\) is an inner generalized derivation;

(ii):

there exist \(a, b \in Q_r\) and if \(b \in {\mathcal {C}}\) then \(eq^m \in {\mathcal {C}}~\text {where}~q=a+b\), provided \({\mathcal {K}}\) is an inner generalized derivation and \(\Re \) satisfies \(s_4\);

(iii):

there exists \(a \in Q_r\) with \(ea=0\), provided \({\mathcal {K}}\) is not an inner generalized derivation;

(iv):

there exists \(a \in Q_r\) with \(ea^m \in {\mathcal {C}}\), provided \({\mathcal {K}}\) is not an inner generalized derivation and \(\Re \) satisfies \(s_4\).