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On Jordan ideals and derivations in rings with involution. (English) Zbl 1211.16037

In the paper, derivations acting as homomorphisms on Jordan ideals in rings with involution are investigated. Namely, the following result is proved: Let \(R\) be a 2-torsion free (i.e., \(2x=0\) implies \(x=0\)) *-prime ring (i.e., for an involution * on \(R\), \(aRb=0=aRb^*\) implies \(a=0\) or \(b=0\)), \(d\) a derivation which commutes with * (i.e., a mapping of \(R\) into \(R\) such that \(d(x+y)=d(x)+d(y)\), \(d(xy)=d(x)y+xd(y)\) and \(d(x^*)=(d(x))^*\) for all \(x,y\in R\)) and \(J\) a non-zero *-Jordan ideal (i.e., an additive subgroup of \(R\) with \(J^*=J\) and \(ur+ru\in J\) for all \(u\in J\) and \(r\in R\)) and a subring of \(R\). If \(d\) acts as a homomorphism (resp. as an antihomomorphism) on \(J\) (i.e., \(d(xy)=d(x)d(y)\) (resp. \(d(xy)=d(y)d(x)\)) for all \(x,y\in J\)), then \(d=0\) or \(J\subseteq Z(R)\).

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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