On Traces of n-Additive Mappings on Semiprime Ring

Abstract

 Let R be a ring with centre Z(R). In this paper we prove that a nonzero Lie ideal L of a semiprime ring R of characteristic different from (2ⁿ-2) is central if it satisfies one of the following: (i)f(xy)∓[x,y]∈Z(R), (ii)f(xy)∓[y,x]∈Z(R), (iii)f(xy)∓xy∈Z(R), (iv)f(xy)∓yx∈Z(R), (v)f([x,y] )∓[x,y]∈Z(R), (vi)f([x,y] )∓[y,x]∈Z(R), (vii)f([x,y] )∓xy∈Z(R),(viii)f([x,y] )∓yx∈Z(R),(ix)f(xy)∓f(x)∓[x,y]∈Z(R),(x)f(xy)∓f(y)∓[x,y]∈Z(R),(xi)f([x,y] )∓f(x)∓[y,x] ∈ Z(R), (xii)f([x,y] )∓f(x)∓[y,x]∈Z(R), (xiii)f([x,y] )∓f(y)∓[x,y]∈Z(R), (xiv)f([x,y] )∓f(y)∓[y,x]∈Z(R), (xv)f([x,y] )∓f(xy)∓[x,y]∈Z(R), (xvi)f([x,y] )∓f(xy)∓[y,x]∈Z(R), (xvii)f(x)f(y)∓[x,y]∈Z(R), (xviii)f(x)f(y)∓[y,x]∈Z(R), (xix)f(x)f(y)∓xy∈Z(R), (xx)f(x)f(y)∓yx∈Z(R) for all x,y∈L, where f stands for the trace of an n-additive map