Nonlinear mappings on full matrices derivable at zero point

Abstract

Let M(n, F) (resp., T(n, F)) be the set of all matrices (resp., upper triangular matrices) over a field F. A mapping on M(n, F) is called derivable at zero point if whenever for . Recently, Wong et al. [Linear Algebra. Appl. 483;2015:236–248] determined all nonlinear mappings (without linear or additive condition) on T(n, F) derivable at zero point. However, a more natural problem is left open: How about the nonlinear mappings on M(n, F) which are derivable at zero point? Let denote the solution space of the homogeneous linear equations with as coefficient matrix. In this article, we solve this problem, proving that a mapping on M(n, F), with , is derivable at zero point if and only if there is and an additive derivation of F such that

where satisfies and denotes the transpose of x. Besides, the problem for the case when is also solved.

Keywords:

• Derivations of rings
• all-derivable points
• full matrices

AMS Subject Classifications:

• 15A03
• 15A04
• 15A27
• 15A99

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grand number 11571360] and by Natural Science Foundation of Anhui Provincial Education Department [grand number EJ2014A009].