Let R be a ring and

be n-additive mapping. A map

is said to be the trace of D if

for all

. Suppose that

are endomorphisms of R. For any

, let < a, b >

. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x),

>

, for all

or

d(x), x >

,

>

, for all

. Further, if < d(x), x >

, the center of R, for all

or < d(x)x - xd(x), x >= 0, for all

, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.