# zbMATH — the first resource for mathematics

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 $$S_4$$.

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