On some additive mappings in rings with involution |
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Authors: | M Bre?ar J Vukman |
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Institution: | (1) Institute of Mathematics, Physics and Mechanics, P.O. Box 543, Y-61001 Ljubljana, Yugoslavia;(2) University of Maribor, VEK , Razlogova 14, Y-62000 Maribor, Yugoslavia |
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Abstract: | 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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Keywords: | Primary 16A72 Secondary 16A12 16A28 16A68 16A70 |
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