Abstract
SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x 1, …,x n) be a polynomial overC in noncommuting variablesx 1, …,x n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a 1, …,a n):a 1, …,a n ∈I}. Then eitherp(x 1, …,x n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.
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Chuang, C.L. The additive subgroup generated by a polynomial. Israel J. Math. 59, 98–106 (1987). https://doi.org/10.1007/BF02779669
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DOI: https://doi.org/10.1007/BF02779669