Summary
LetR be a *-ring. We study an additive mappingD: R → R satisfyingD(x 2) =D(x)x * +xD(x) for allx ∈ R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana ∈ R such thatD(x) = ax * − xa for allx ∈ R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx * =x * x for allx ∈ R) if and only if there exists a nonzero additive mappingD: R → R satisfyingD(x 2) =D(x)x * +xD(x) and [D(x), x] = 0 for allx ∈ R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx ∈ R, forcesR to be commutative.
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Brešar, M., Vukman, J. On some additive mappings in rings with involution. Aeq. Math. 38, 178–185 (1989). https://doi.org/10.1007/BF01840003
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DOI: https://doi.org/10.1007/BF01840003