Abstract
The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.
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REFERENCES
K. I. Beidar and S. A. Pikhtilkov, “The prime radical of special Lie algebras,” Fundam. Prikl. Mat. [in Russian], 6 (2000), no. 3, 643-648.
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow, 1996.
Yu. I. Merzlyakov, Rational Groups [in Russian], Nauka, Moscow, 1980.
G. Baumslag, L. G. Kovacs, and B. H. Neumann, “On products of normal subgroups,” Acta Sci. Math., 26 (1965), 145-147.
K. I. Shchukin, “The RI * solvable radical of groups,” Mat. Sb. [Math. USSR-Sb.], 52 (1960), no. 4, 1021-1031.
A. Buys and G. K. Gerber, “The prime radical for Ω-groups,” Comm. Algebra, 10 (1982), 1089-1099.
I. Z. Golubchik and A. V. Mikhalev, “Isomorphisms of unitary groups over associative rings,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) [in Russian], 132 (1983), 97-109.
A. V. Mikhalev and M. A. Shatalova, “The prime radical of Ω-groups and Ω-l-groups,” Fundam. Prikl. Mat. [in Russian], 4 (1998), no. 4, 1405-1413.
A. Yu. Golubkov, “The prime (RI*-decidable) radical of the unitary group over a ring with involution,” Fundam. Prikl. Mat. [in Russian], 6 (2000), no. 1, 93-119.
Zhirang Zhang, “The locally solvable radical of locally finite groups,” Southeast Asian Bull. Math., 22 (1998), no. 3, 325-329.
Zhirang Zhang and Wenze Yang, “The locally solvable radical of torsion groups,” Comm. Algebra, 28 (2000), no. 8, 3987-3991.
N. Jacobson, PI-Algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
Yu. P. Razmyslov, Identities of Algebras and Their Representations [in Russian], Nauka, Moscow, 1989; English transl. in: Translations of Mathematical Monographs, 138, American Mathematical Society, Providence, RI, 1994.
A. R. Kemer, “Capelli identities and nilpotency of the radical of finitely generated PI algebra,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 245 (1980), no. 4, 793-797.
A. Braun, “The nilpotency of the radical in a finitely generated PI-ring,” J. Algebra, 89 (1984), no. 2, 375-396.
I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, no. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968.
J. Lambek, Lectures on Rings and Modules, Second edition, Chelsea Publishing Co., New York, 1976.
D. S. Passman, “Group rings satisfying a polynomial identity,” J. Algebra, 20 (1972), 103-117.
Yu. P. Razmyslov, “Lie algebras satisfying Engel conditions,” Algebra i Logika [Algebra and Logic], 10 (1971), no. 1, 33-44.
Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1976.
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Pikhtil'kov, S.A. On the Prime Radical of PI-Representable Groups. Mathematical Notes 72, 682–686 (2002). https://doi.org/10.1023/A:1021465207727
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DOI: https://doi.org/10.1023/A:1021465207727