**Authors:** Beidar K. I., Wiegandt R.

### Abstract

The radical class of Passman's radical assignment is the Levitzki radical class, this is used to charactarize the Levitzki radical class and its semisimple class. Locally $f$-solvable rings are defined in terms of a polynomial in noncommutative variable with integral coefficients and without linear terms, as a generalization of PI rings. Locally $f$-solvable rings form a special radical class, and examples are the Levitzki and K\"othe's nil radical. Under certain condition on $f$, the structure of primitive rings with nonzero locally $f$-solvable radical is described, in particular, a locally $f$-solvable ring modulo its K\"othe radical satisfies a standard identity. On the class of locally $f$-solvable rings the K\"othe problem has a positive solution. Designating to an alternative ring its associator ideal is a complete Hoehnke radical with Kurosh-Amitsur radical class. Alternative rings with associative semiprime factor rings form a Kurosh-Amitsur radical class which is the upper radical of all nonassociative alternative prime rings.

Departament of Mathematics, National Cheng Kung University

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Mathematical Institute HAS,

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