The objective of this paper is to study Tri-additive maps and permuting Tri-derivations of G-rings. The main result of the paper is written as follows. Let M be a 2, 3-torsion free left -unital G-ring and D be a permuting Tri-additive mapping on M with the trace d.

·         If d is skew-commuting on M, then  and d is commutative on M.

·         If  for all  then  or  for a fixed element a of M.

·         Let    and  be traces of permuting Tri-derivations  and  respectively. If the identity  where  and  holds, then there exists an element p of the extended centroid of M such that  for all