Centralizing mappings of semiprime rings. (English) Zbl 0614.16026

Let \(R\) be a semi-prime ring, \(Z\) the center of \(R\), \(L\) a nonzero left ideal of \(R\), and \(T\) an endomorphism of \(R\). Call \(T\) centralizing on \(L\) if \([x,T(x)]\in Z\) for all \(x\in L\). The first main result of the paper is that if \(T\) is one-one and centralizing on \(L\), if \(U=L\cap T^{-1}(L)\cap T^{- 2}(L)\cap T^{-3}(L)\neq 0\), and if \(T\) is not the identity map on \(U\), then \(R\) contains a nonzero central ideal. The authors also show that when \(R\) is a prime ring, \(T\) is not the identity map, and \(T\) is one-one and centralizing on \(L\), then \(R\) is commutative. The last main result generalizes a theorem of E. C. Posner [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] by proving that if \(D\) is a derivation of \(R\) which is centralizing on \(L\) then either \(D(L)=0\) or \(R\) contains a nonzero central ideal.
Reviewer: C.Lanski


16W20 Automorphisms and endomorphisms
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16N60 Prime and semiprime associative rings
16Dxx Modules, bimodules and ideals in associative algebras


Zbl 0082.03003
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