Di Vincenzo, O. M. On the \(n\)-th centralizer of a Lie ideal. (English) Zbl 0692.16022 Boll. Unione Mat. Ital., VII. Ser., A 3, No. 1, 77-85 (1989). Let R be a ring. The nth centralizer \(C_ n(T)\) of a subset T of R is the set of elements \(a\in R\) satisfying \(S_ n(a,t_ 2,...,t_ n)=0,\quad for\quad all\quad t_ i\in T\) \((i=2,...,n)\), where \(S_ n(x_ 1,...,x_ n)=\sum (-1)^{\sigma}x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\) and the sum is over all permutations \(\sigma\) of the non-commuting unknowns \(x_ 1,...,x_ n\). A. Kovacs [J. Algebra 40, 107-124 (1976; Zbl 0329.16010)] has shown that if R is a prime ring then either \(C_{2n}(R)=R\) or \(C_{2n}(R)=Z(R)\), the centre of R, and \(C_{2n+1}(R)=R\) or \(C_{2n+1}(R)=0\). In the present paper the author investigates the case in which T is a Lie ideal U of R; \(C_ n(U)\) is then also a Lie ideal of R. An example in which R is the ring of \(2\times 2\) matrices over a field of characteristic 2 shows that the results of Kovacs do not extend in full to this case, but the following results are established: If U is a non-commutative Lie ideal of a prime ring R then (i) either \(C_{2n}(U)=R\) and R satisfies \(S_{2n}(x_ 1,...,x_{2n}),\) or \(C_{2n}(U)=Z(R)\) and (ii) either \(C_{2n+1}(U)=R\) and R satisfies \(S_{2n+1}(x_ 1,...,x_{2n+1}),\) or \(C_{2n+1}(U)=0\); except in the case \(n=3\) when R has characteristic 3 and R is an order in a 4-dimensional central simple algebra. When U is commutative either \(U\subseteq Z(R)\) and \(C_ n(U)=R\) or \(U\not\subseteq Z(R)\), the characteristic of R is 2 and R satisfies \(S_ 4(x_ 1,x_ 2,x_ 3,x_ 4)\). Reviewer: E.M.Patterson Cited in 39 Documents MSC: 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16Rxx Rings with polynomial identity 16N60 Prime and semiprime associative rings 16Dxx Modules, bimodules and ideals in associative algebras 16U70 Center, normalizer (invariant elements) (associative rings and algebras) Keywords:centralizer; prime ring; centre; non-commutative Lie ideal Citations:Zbl 0329.16010 PDFBibTeX XMLCite \textit{O. M. Di Vincenzo}, Boll. Unione Mat. Ital., VII. Ser., A 3, No. 1, 77--85 (1989; Zbl 0692.16022)