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)].


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)


Zbl 0746.16029
Full Text: DOI EuDML