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

期刊论文
发表日期：

2024年2月22日

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Mohammad Aslam Siddeeque,
Ali Ahmed Abdullah,
Nazim Khan

摘要

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

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

(ii)

there exist $a, b \in Q_r$ $a, b \in Q_{r}$ and if $b \in {\mathcal {C}}$ $b \in C$ then $eq^m \in {\mathcal {C}}~\text {where}~q=a+b$ $e q^{m} \in C where q = a + b$ , provided ${\mathcal {K}}$ $K$ is an inner generalized derivation and $\Re $ $ℜ$ satisfies $s_4$ $s_{4}$ ;

(iii)

there exists $a \in Q_r$ $a \in Q_{r}$ with $ea=0$ $e a = 0$ , provided ${\mathcal {K}}$ $K$ is not an inner generalized derivation;

(iv)

there exists $a \in Q_r$ $a \in Q_{r}$ with $ea^m \in {\mathcal {C}}$ $e a^{m} \in C$ , provided ${\mathcal {K}}$ $K$ is not an inner generalized derivation and $\Re $ $ℜ$ satisfies $s_4$ $s_{4}$ .