对于任意给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}={{x,y}k1,y}1,其中{x,y}0=x,{x,y}1=xy+yx.假设R是含有单位元与非平凡幂等元的环,f:R→R是满射.文章证明了在一定的假设条件下,f满足{f(x),f(y)}k={x,y}k对所有的x,y∈R成立当且仅当f(x)=λx对所有的x∈R成立,其中λ∈L(R)(R的中心)且λk+ 1=1.作为应用,给出了素环与von Neumann代数上保持此类性质映射的完全刻画.%For any integer k≥1,the k-Jordan product of two elements x,y in a ring R is defined by {x,y}k ={{x,y}k-1,y}1,where {x,y}0=xand {x,y}1=xy+-yx.Assume that Risa unital ring containing a non-trivial idempotent and f:R→R is a surjective map.It is shown that,under some mild conditions,f satisfies {f(x),f(y)}k={x,y}k for all x,y∈R if and only if there exists λ∈Z(R) (the center of R) with λk+1 =1 such that f(x)=λx holds for all x ∈R.As an application,such maps on prime rings and von Neumann algebras are characterized,respectively.
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