Let A and B be unital C-algebras and h: A B be bijective map such that h(0) = 0, and for all n=0,1,2, , for all x A, and for all u U(A), h(2nuy) = h(2nu)h(y). At this dissertation, we show that under some conditions, h: A B is a C-algebra isomorphism and by enforcing real rank zero condition on a C-algebra A and weakening some others, the assertion still remains true. At the end, by using Ulam-Mazur theorem, we give a generalization of this problem for isometries between left normed modules