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An extension to a theorem of Hua. (English) Zbl 1399.12006

Summary: 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 \(f(a\cdot b\cdot c) = f(a) \cdot f(b) \cdot f(c)\) or else \(f(a\cdot b\cdot c) = f(c) \cdot f(b) \cdot f(a)\) holds. As a consequence, the famous theorem of L. K. Hua given in 1949 [Proc. Natl. Acad. Sci. USA 35, 386–389 (1949; Zbl 0033.10402)] 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.

MSC:

12E15 Skew fields, division rings
16K40 Infinite-dimensional and general division rings
16W20 Automorphisms and endomorphisms

Citations:

Zbl 0033.10402
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