Multiplicative maps of matrices over rings

Abstract

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring $M_n(R)$ is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

Keywords:

• Additivity
• multiplicative isomorphism
• ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W20
• 15A86
• 47L10

Acknowledgments

The author is grateful to the referee for detailed comments and many valuable suggestions. He wishes to thank Professor Yanan Lin for his encouragement. He would like to thank Zhankui Xiao for many helpful discussions.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 11701486) and the Natural Science Foundation of Fujian Province of China (No. 2020J05234).