Additive maps of semiprime rings satisfying an Engel condition
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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 $x\in R$.
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 $x\in R$.
참고문헌 (27)
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