Commuting anti-homomorphisms

ABSTRACT

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) $x{x}^{\tau }=x^{\tau }x$ for all x∈R; (2) x+x^τ∈Z(R) for all x∈R; (3) xx^τ∈Z(R) for all x∈R. In this case, there exists an idempotent e∈C such that (1−e)R is a commutative ring and the semiprime ring eR is equipped with an involution $\tilde{\tau }$, which is induced canonically by τ. These results generalize both the involution case by Chacron [5] and the anti-automorphism case by the author [7].

KEYWORDS:

• Anti-automorphism
• anti-homomorphism
• central trace (norm) ring
• commuting map
• involution
• linear identity
• semiprime ring
• small centralizer

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W10
• 16W20
• 16R50
• 16R60

Acknowledgments

The author is grateful to the referee for the valuable suggestions which help to clarify the paper. The work was supported in part by the Ministry of Science and Technology of Taiwan (MOST 105-2115-M-002 -003 -MY2) and the National Center for Theoretical Sciences (NCTS), Taipei Office.