Symmetric $n$-Additive Mappings Admitting Semiprime Ring

Kapil Kumar

Kragujevac Journal of Mathematics, Tome 49 (2025) no. 5, p. 755

Résumé

Let $\mathcal{R}$ be a ring with centre $Z(\mathcal{R})$. An $n$-additive map $D:\mathcal{R}^{n}\rightarrow \mathcal{R}$ is called symmetric $n$-additive if $D(x_{1},…,x_{n})=D(x_{\pi(1)},…,x_{\pi(n)})~\mbox{for all}~ x_{i}\in \mathcal{R}$ and for every permutation ${({\pi(1)},{\pi(2)},…,{\pi(n)})}$. A mapping $\triangle:\mathcal{R}\rightarrow \mathcal{R}$ defined by $\triangle(x)=D(x,x,…,x)$ is called the trace of $D$. In this paper, we prove that a nonzero Lie ideal $L$ of a semiprime ring $\mathcal{R}$ of characteristic different from $(2^n-2)$ is central, if it satisfies any one of the following properties: (i) $\triangle([x,y])\mp xy\in Z(\mathcal{R})$; (ii) $\triangle([x,y])\mp [y,x]\in Z(\mathcal{R})$; (iii) $\triangle(xy)\mp \triangle(x)\mp [x,y] \in Z(\mathcal{R})$; (iv) $\triangle([x,y])\mp yx\in Z(\mathcal{R})$; (v) $\triangle(xy)\mp \triangle(y)\mp [x,y] \in Z(\mathcal{R})$.

Détail

Classification : 16W25, 16R50, 16N60
Keywords: semiprime rings, Lie ideals, symmetric n-additive mapping, trace.