## Rocky Mountain Journal of Mathematics

### Strong commutativity preserving maps on rings

#### Abstract

Suppose $\R$ is a unital ring having an idempotent element $e$ which satisfies $a{\R}e=0$ implies $a=0$ and $a{\R}(1-e)=0$ implies $a=0$. In this paper, we aim to characterize the map $f:{\R}\rightarrow {\R}$, $f$ is surjective and $[f(x),f(y)]=[x,y]$ for all $x,y\in {\R}$. It is shown that $f(x)=\alpha x +\xi (x)$ for all $x\in {\R}$, where $\alpha \in {\Z}({\R})$, $\alpha^2=1$, and $\xi$ is a map from ${\R}$ into ${\Z}({\R})$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $I_1$ is obtained.

#### Article information

Source
Rocky Mountain J. Math., Volume 44, Number 3 (2014), 733-742.

Dates
First available in Project Euclid: 28 September 2014

https://projecteuclid.org/euclid.rmjm/1411945661

Digital Object Identifier
doi:10.1216/RMJ-2014-44-3-733

Mathematical Reviews number (MathSciNet)
MR3264479

Zentralblatt MATH identifier
1312.16041

Subjects
Primary: 16N60. 16U80: Generalizations of commutativity

#### Citation

Bai, Zhaofang; Du, Shuanping. Strong commutativity preserving maps on rings. Rocky Mountain J. Math. 44 (2014), no. 3, 733--742. doi:10.1216/RMJ-2014-44-3-733. https://projecteuclid.org/euclid.rmjm/1411945661

#### References

• F.W. Anderson and K.R. Fuller, Rings and categories of modules, Springer-Verlag, New York, 2004.
• Z.F. Bai and S.P. Du, Multiplicative $*$-Lie isomorphism between factors, J. Math. Anal. Appl. \bf346 (2008), 327-335.
• Z.F. Bai and S.P. Du, Multiplicative $*$-Lie isomorphism between von Neumann algebras, Linear Multilinear Alg. \bf60 (2012), 311-322.
• –––, Maps preserving products $XY-YX^*$ on von Neumann algebras, J. Math. Anal. Appl. \bf386 (2012), 103-109.
• Z.F. Bai, S.P. Du and J.C. Hou, Multiplicative Lie isomorphism between prime rings, Comm. Alg. \bf36 (2008), 1626-1633.
• R. Banning and M. Mathieu, Commutativity preserving mappings on semiprime rings, Comm. Alg. \bf25 (1997), 247-265.
• H.E. Bell and M.N. Daif, On commutativity and strong commutativity preserving maps, Canad. Math. Bull. \bf37 (1994), 443-447.
• M. Brešar and C.R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. \bf37 (1994), 457-460.
• M. Brešar and P. Šemrl, Commutativity preserving linear maps on central simple algebras, J. Alg. \bf284 (2005), 102-110.
• M.D. Choi, A.A. Jafarian and H. Radjavi, Linear maps preserving commutativity, Linear Alg. Appl. \bf87 (1987), 227-242.
• Q. Deng and M. Ashraf, On strong commutativity preserving maps, Results Math. \bf30 (1996), 259-263.
• M.D. Dolinar, S.P. Du, J.C. Hou and P. Legiša, General preservers of invariant subspace lattices, Linear Alg. Appl. \bf429 (2008), 100-109.
• R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras, Vol. I, Academic Press, New York, 1983; Vol. II, Academic Press, New York, 1986.
• Jer-Shyong Lin and Cheng-Kai Liu, Strong commutativity preserving maps on Lie ideals, Linear Alg. Appl. \bf428 (2008), 1601-1609.
• –––, Strong commutativity preserving maps in prime rings with involution, Linear Alg. Appl. \bf432 (2010), 14-23.
• C.R. Miers, Lie isomorphisms of operator algebras, Pacific J. Math. \bf38 (1971), 717-735.
• L. Molnar, Selected preserver problems on algebraic structures of linear operators and on function spaces, Springer-Verlag, Berlin, 2007.
• L. Molnár and P. Šemrl, Non-linear commutativity preserving maps on self-adjoint operators, Quart. J. Math. \bf56 (2005), 589-595.
• X.F. Qi and J. Hou, Nonlinear strong commutativity preserving maps on prime rings, Comm. Alg. \bf38 (2010), 2790-2796.
• P. Šemrl, Non-linear commutativity preserving maps, Acta Sci. Math. \bf71 (2005), 781-819. \noindentstyle