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On the \(n\)-th centralizer of a Lie ideal. (English) Zbl 0692.16022

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

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)

Citations:

Zbl 0329.16010
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