$k$-power centralizing and $k$-power skew-centralizing maps on‎ ‎triangular rings

Document Type : Research Paper

Author

Department of Mathematics‎, ‎Shanxi University‎, ‎Taiyuan 030006‎, ‎P‎. ‎R‎. ‎China.

Abstract

‎Let $\mathcal A$ and $\mathcal B$ be unital rings‎, ‎and $\mathcal M$‎ ‎be an $(\mathcal A‎, ‎\mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $\mathcal A$-module and also as a right $\mathcal B$-module‎.  ‎Let ${\mathcal U}=Tri(\mathcal A‎, ‎\mathcal M‎, ‎\mathcal‎ ‎B)$ be the triangular ring and ${\mathcal Z}({\mathcal U})$ its‎  ‎center‎. ‎Assume that $f:{\mathcal U}\rightarrow{\mathcal U}$ is a map‎  ‎satisfying $f(x+y)-f(x)-f(y)\in{\mathcal Z}({\mathcal U})$ for all‎ ‎$x,\ y\in{\mathcal U}$ and $k$ is a positive integer‎. ‎It is shown‎  ‎that‎, ‎under some mild conditions‎, ‎the following statements are‎  ‎equivalent‎: ‎(1) $[f(x),x^k]\in{\mathcal Z}({\mathcal U})$ for all‎  ‎$x\in{\mathcal U}$; (2)  $[f(x),x^k]=0$ for all $x\in{\mathcal U}$;‎ ‎(3) $[f(x),x]=0$ for all $x\in{\mathcal U}$; (4) there exist a‎  ‎central element $z\in{\mathcal Z}({\mathcal U})$ and an additive‎ ‎modulo ${\mathcal Z}({\mathcal U})$ map $h:{\mathcal‎
‎U}\rightarrow{\mathcal Z}({\mathcal U})$ such that $f(x)=zx+h(x)$‎ ‎for all $x\in{\mathcal U}$‎. ‎It is also shown that there is no‎ ‎nonzero additive $k$-skew-centralizing maps on triangular rings.
 

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H. E. Bell and J. Lucier, On additive maps and commutativity in rings, Result Math.36 (1999), no. 1-2, 1--8.
M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111 (1991), no. 2, 501--510.
M. Brešar, On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47 (1993), no. 2, 291--296.
M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), no. 2, 385--394.
M. Brešar, On generalized biderivations and related maps, J. Algebra 172 (1995), no. 3, 764--786.
M. Brešar, Commuting maps: a survey, Taiwanese J. Math. 8 (2004), no. 3, 361--397.
M. Brešar and B. Hvala, On additive maps of prime rings, Bull. Austral. Math. Soc. 51 (1995), no. 3, 377--381.
C. W. Chen, M. T. Koşan and T. K. Lee, Decompositions of quotient rings and m-power commuting maps, Comm. Algebra 41 (2011), no. 5, 1865--1871.
W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc. (2) 63 (2001), no. 1, 117--127.
K. R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics, John Wiley & Sons, Inc., New York, 1988
N. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canad. Sect. III (3) 49 (1955) 19--22.
Y. Q. Du and Y.Wang, k--commuting maps on triangular algebras, Linear Algebra Appl. 436 (2012), no. 5, 1367--1375.
H. G. Inceboz, M. T. Koşan and T. K. Lee, m-power commuting maps on semiprime rings, Comm. Algebra 42 (2014), no. 3, 1095--1110.
C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993) 75--80.
C. Lanski, An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc. 125 (1997), no. 2, 339--345.
X. F. Qi and J. C. Hou, Characterization of k-commuting additive maps on rings, Linear Algebra Appl. 468 (2015) 48--62.
J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47--52.