Additive maps of semiprime rings satisfying an Engel condition
Bull. Korean Math. Soc. 2021 Vol. 58, No. 3, 659-668
Published online January 7, 2021
Printed May 31, 2021
Tsiu-Kwen Lee, Yu Li, Gaohua Tang
National Taiwan University; Southwest University; Beibu Gulf University
Abstract : Let $R$ be a semiprime ring with maximal right ring of quotients $Q_{mr}(R)$, and let $n_1, n_2,\ldots ,n_k$ be $k$ fixed positive integers. Suppose that $R$ is $\big(n_1+ n_2+\cdots+n_k\big)!$-torsion free, and that $f\colon \rho\to Q_{mr}(R)$ is an additive map, where $\rho$ is a nonzero right ideal of $R$. It is proved that if $\Big[\big[\ldots [f(x), x^{n_1}],\ldots\big], x^{n_k}\Big]=0$ for all $x\in \rho$, then $\big[f(x), x\big]=0$ for all $x\in \rho$. This gives the result of Beidar et al. \cite{beidar1997} for semiprime rings. Moreover, it is also proved that if $R$ is $p$-torsion, where $p$ is a prime integer with $p=\sum_{i=1}^kn_i$, and if $f\colon R\to Q_{mr}(R)$ is an additive map satisfying $\Big[\big[\ldots [f(x), x^{n_1}],\ldots\big], x^{n_k}\Big]=0$ for all $x\in R$, then $\big[f(x), x\big]=0$ for all $x\in R$.
Keywords : Semiprime ring, prime ring, extended centroid, Engel condition, functional identity
MSC numbers : Primary 16R60, 16N60
Supported by : The work of G. H. Tang was supported by the national natural science foundation of China (11661014, 11961050, 11661013), the work of T.-K. Lee was supported in part by the Ministry of Science and Technology of Taiwan (MOST 107-2115-M-002-018-MY2)
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