On centralizers of semiprime rings. (English) Zbl 0746.16011

Zalar, Borut

Commentat. Math. Univ. Carol. 32, No. 4, 609-614 (1991).

The author proves two results on mappings of a semi-prime 2-torsion free ring \(R\). An additive map \(T\colon R\to R\) is a left (right) centralizer if \(T(xy)=T(x)y\) (\(T(xy)=xT(y)\)) for all \(x,y\in R\), and is a Jordan centralizer if \(T(xy+yx)=T(x)y+yT(x)=xT(y)+T(y)x\) for all \(x,y\in R\). The first main result proves that if \(T(x^2)=T(x)x\) for all \(x\in R\), then \(T\) is a left centralizer. The second shows that any Jordan centralizer of \(R\) is both a left and right centralizer.

Reviewer: C.Lanski (Los Angeles)

MSC:

16N60	Prime and semiprime associative rings
16W10	Rings with involution; Lie, Jordan and other nonassociative structures
16W20	Automorphisms and endomorphisms
16U70	Center, normalizer (invariant elements) (associative rings and algebras)

Keywords:

semi-prime free rings; additive maps; left centralizers; Jordan centralizers; right centralizers