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Power reduction property for generalized identities of one-sided ideals. (English) Zbl 0845.16017
Let \(R\) be a semiprime ring, \(U\) be the left Utumi quotient of \(R\). Suppose that a left ideal \(I\) of \(R\) satisfies the GPI \(f(x_1^{n_1},\dots,x^{n_m}_m)=0\), where \(f(x_1,\dots,x_m)\) is a multilinear generalized polynomial with coefficients in \(U\). The author proves that \(I\) satisfies the identity \(f(x_1,\dots,x_m)=0\). As a corollary the author obtains the following generalization of Felzenszwalb’s result: Let \(R\) be a prime ring, \(I<_eR\), \(C\) the extended centroid of \(R\) and \(d\) a nonzero derivation of \(R\) such that for some multilinear polynomial \(f(x_1,\dots,x_m)\) over \(C\) \(d(f(x_1^{n_1},\dots,x^{n_m}_m))\in C\), where \(n_1,\dots,n_m\) are fixed numbers. Then either \(f(x_1,\dots,x_m)\) is central-valued on \(I/I\cap r_R(I)\) or \(\text{char }R=2\) and \(I/I\cap r_R(R)\) satisfies \(S_4\).

16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras