Abstract: Let N be zero-symmetric prime near-ring and Z the center of N,U be a nonzero ideal of N.It is shown that if T is a nontrivial automorphism of N such that u,T(u)∈Z,and T(u) is in U for every u in U,and if U is distributive,then N is a commutive prime ring.And if N is a 2-torision free distributive prime near-ring,then U need only be a nonzero Jordan ideal.