Commuting anti-homomorphisms. Issue 3 (4th March 2018)

• Record Type: Journal Article
• Authors: Lee, Tsiu-Kwen
• Is Part Of: Communications in algebra. Volume 46:Issue 3(2018)
• Page Start: 1060
• Page End: 1065
• Publication Date: 2018-03-04
• Subjects: Anti-automorphism -- anti-homomorphism -- central trace (norm) ring -- commuting map -- involution -- linear identity -- semiprime ring -- small centralizer; 16N60 -- 16W10 -- 16W20 -- 16R50 -- 16R60; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1335746
• Languages: English
• ISSNs: 0092-7872

Abstract

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 τ = x τ x for all x ∈ R ; (2) x + x τ ∈ Z ( R ) for all x ∈ R ; (3) x x 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 involutionτ ̃, 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 ].