An Extension to a Theorem of Hua

Let f be an injective map of a division ring D to itself. Suppose that the sum of the images of elements from D is equal to the image of the sum of those elements under consideration, and that for each non-zero element s from D, the unordered product of the images of s and s−1 equals the unit element 1 of D. In this article, we will show that for any elements a, b, c from D, either we have f(a·b·c) = f(a)·f(b)·f(c) or else f(a · b · c) = f(c) · f(b) · f(a) holds. As a consequence, the famous theorem of L.K Hua given in 1949 has been substantially expanded. Some other consequences are stated and proved,while related results to Hua’s theorem and results of others are also discussed and elucidated.