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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)
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References:

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