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A note on semiprime rings with derivation. (English) Zbl 0879.16025

Let \(R\) denote a 2-torsion-free semiprime ring with center \(Z\), and \(I\) a nonzero ideal of \(R\). It is proved that \(I\subseteq Z\) if and only if \(R\) admits a derivation \(d\) having one of the following properties: (i) \(d([x,y])-[x,y]\in Z\) for all \(x,y\in I\); (ii) \(d([x,y])+[x,y]\in Z\) for all \(x,y\in I\); (iii) for each \(x,y\in I\), \(d([x,y])-[x,y]\in Z\) or \(d([x,y])+[x,y]\in Z\). This result generalizes earlier work of M. N. Daif and H. E. Bell [Int. J. Math. Math. Sci. 15, No. 1, 205-206 (1992; Zbl 0746.16029)].

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)

Citations:

Zbl 0746.16029
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