IMCS/Publications/BASM/Issues/BASM n.2 (30), 1999/

Radical assignments and radical classes. (English)

Authors: Beidar K. I., Wiegandt R.


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
Tainan 70101. Taiwan, R.O.C
Mathematical Institute HAS,
P.O. Box 127, H-1364 Budapest, Hungary