NONADDITIVE STRONG COMMUTATIVITY PRESERVING DERIVATIONS AND ENDOMORPHISMS

Title & Authors
NONADDITIVE STRONG COMMUTATIVITY PRESERVING DERIVATIONS AND ENDOMORPHISMS
Zhang, Wei; Xu, Xiaowei;

Abstract
Let S be a nonempty subset of a ring R. A map $\small{f:R{\rightarrow}R}$ is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all $\small{x,y{\in}S}$, where the symbol [x, y] denotes xy - yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal $\small{{\rho}}$ of R, then $\small{{\rho}{\subseteq}Z}$, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal $\small{I{\cup}T^{-1}(I)}$, then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.
Keywords
semiprime ring;prime ring;strong commutativity preserving map;
Language
English
Cited by
References
1.
S. Ali and S. Huang, On derivations in semiprime rings, Algebr. Represent. Theory 15 (2012), no. 6, 1023-1033.

2.
K. T. Beidar, W. S. Martindale, and A. V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics 196, Marcel Dekker, New York, 1996.

3.
H. E. Bell and M. N. Daif, On commutativity and strong commutativity preserving maps, Canad. Math. Bull. 37 (1994), no. 4, 443-447.

4.
H. E. Bell and G. Mason, On derivations in near rings and rings, Math. J. Okayama Univ. 34 (1992), 135-144.

5.
M. Bresar, and C. R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. 37 (1994), no. 4, 457-460.

6.
V. De Filippis and G. Scudo, Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear Multilinear Algebra 61 (2013), 917-938.

7.
Q. Deng and M. Ashraf, On strong commutativity preserving mappings, Results Math. 30 (1996), no. 3-4, 259-263.

8.
T. K. Lee and T. L. Wong, Nonadditive strong commutativity preserving maps, Comm. Algebra 40 (2012), no. 6, 2213-2218.

9.
P. K. Liau, W. L. Huang, and C. K. Liu, Nonlinear strong commutativity preserving maps on skew elements of prime rings with involution, Linear Algebra Appl. 436 (2012), no. 9, 3099-3108.

10.
J. S. Lin and C. K. Liu, Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008), no. 7, 1601-1609.

11.
C. K. Liu, Strong commutativity preserving generalized derivations on right ideals, Monatsh. Math. 166 (2012), no. 3-4, 453-465.

12.
C. K. Liu and P. K. Liau, Strong commutativity preserving generalized derivations on Lie ideals, Linear Multilinear Algebra, 59 (2011), no. 8, 905-915.

13.
J. Ma, X. W. Xu, and F. W. Niu, Strong commutativity-preserving generalized derivations on semiprime rings, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 11, 1835-1842.

14.
X. F. Qi and J. C. Hou, Nonlinear strong commutativity preserving maps on prime rings, Comm. Algebra 38 (2010), no. 8, 2790-2796.