ON 3-ADDITIVE MAPPINGS AND COMMUTATIVITY IN CERTAIN RINGS

Title & Authors
ON 3-ADDITIVE MAPPINGS AND COMMUTATIVITY IN CERTAIN RINGS
Park, Kyoo-Hong; Jung, Yong-Soo;

Abstract
Let R be a ring with left identity e and suitably-restricted additive torsion, and Z(R) its center. Let H : $\small{R{\times}R{\times}R{\rightarrow}R}$ be a symmetric 3-additive mapping, and let h be the trace of H. In this paper we show that (i) if for each $\small{x{\in}R}$, $$n=<<\cdots,\;x>,\;\cdots,x>{\in}Z(R)$$ with $\small{n\geq1}$ fixed, then h is commuting on R. Moreover, h is of the form $\small{h(x)=\lambda_0x^3+\lambda_1(x)x^2+\lambda_2(x)x+\lambda_3(x)\;for\;all\;x{\in}R}$, where $\small{\lambda_0\;{\in}\;Z(R)}$, $\small{\lambda_1\;:\;R{\rightarrow}R}$ is an additive commuting mapping, $\small{\lambda_2\;:\;R{\rightarrow}R}$ is the commuting trace of a bi-additive mapping and the mapping $\small{\lambda_3\;:\;R{\rightarrow}Z(R)}$ is the trace of a symmetric 3-additive mapping; (ii) for each $\small{x{\in}R}$, either $n=0\;or\;<n,\;x^m>=0$ with $\small{n\geq0,\;m\geq1}$ fixed, then h = 0 on R, where 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.
Keywords
skew-commuting mappings;skew-centralizing mappings;commuting mappings;derivations;
Language
English
Cited by
1.
On Skew Centralizing Traces of Permuting n-Additive Mappings, Kyungpook mathematical journal, 2015, 55, 1, 1
References
1.
H. E. Bell and J. Lucier, On additive maps and commutativity in rings, Result. Math. 36 (1999), 1-8

2.
M. Bresar, Commuting maps: a survey, Taiwanese J. Math. 8 (2004), no. 3, 361-397

3.
Q. Deng and H. E. Bell, On derivations and commutativity in semiprime rings, Comm. Algebra 23 (1995), 3705-3713

4.
K.-H. Park and Y.-S. Jung, Skew-commuting and commuting mappings in rings, Aequa-tiones Math. 64 (2002), 136-144

5.
E. Posner, Derivations in Prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1102