## On some additive mappings in rings with involution.(English)Zbl 0691.16041

Let $$R$$ be a ring with an involution $$x\mapsto x^*$$. The authors study an additive mapping $$D: R\to R$$ satisfying the condition $$D(x^ 2)x^*+xD(x)$$ and prove the following results: i) $$R$$ contains the unit, the element $$1/2$$ and an invertible skew-Hermitian element lying in its centre, implies there exists an $$a\in R$$, such that $$D(x)=ax^*-xa$$ for all $$x\in R$$. ii) $$R$$ is a noncommutative prime real algebra implies $$D$$ is linear, and iii) $$R$$, a noncommutative prime ring of characteristic $$\neq 2$$, is normal ($$xx^*=x^*x$$ for all $$x\in R)\Leftrightarrow\exists$$ a nonzero $$D$$ satisfying $$D(x^ 2)=D(x)x^*+xD(x)$$ and $$[D(x),x]=0$$.
Reviewer: S.A.Huq

### MSC:

 16W20 Automorphisms and endomorphisms 16N60 Prime and semiprime associative rings 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16U70 Center, normalizer (invariant elements) (associative rings and algebras)

### Keywords:

additive mappings; skew-Hermitian elements; prime rings
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### References:

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