On 3-additive mappings and commutativity in certain rings

Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 41-51 Printed March 1, 2007

Kyoo-Hong Park, Yong-Soo Jung Seowon University, Sun Moon University

Abstract : Let $R$ be a ring with left identity $e$ and suitably-restricted additive torsion, and $Z(R)$ its center. Let $H : R \times R \times R \to R$ be a symmetric 3-additive mapping, and let $h$ be the trace of $H$. In this paper we show that (i) if for each $x \in R$, $$\langle h(x),\,x \rangle _{n}=\langle\langle \cdots \langle h(x),\,x \rangle,\,x \rangle, \ldots, x \rangle \in Z(R)$$ with $n\geq1$ fixed, then $h$ is commuting on $R$. Moreover, $h$ is of the form $$ h(x)=\lambda_{0}x^{3}+\lambda_{1}(x)x^{2}+\lambda_{2}(x)x+\lambda_{3}(x) \ \ \mbox{for all} \ x \in R, $$ where $\lambda_{0} \in Z(R)$, $\lambda_{1}:R \to R$ is an additive commuting mapping, $\lambda_{2}:R \to R$ is the commuting trace of a bi-additive mapping and the mapping $\lambda_{3}:R \to Z(R)$ is the trace of a symmetric $3$-additive mapping; (ii) for each $x \in R$, either $\langle h(x),\,x \rangle_{n}=0$ or $\langle\langle h(x),\,x \rangle_{n},\,x^{m} \rangle=0$ with $n\geq0, \ m\geq1$ fixed, then $h=0$ on $R$, where $\langle y, x \rangle$ denotes the product $yx+xy$ and $Z(R)$ is the center of $R$. We also present the conditions which implies commutativity in rings with identity as motivated by the above result.