Abstract
Let R be an Artinian ring, not necessarily with a unit, and let R º be the group of all invertible elements of R with respect to the operation a º b = a + b + ab. We prove that the group R º is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring R by its Jacobson radical is commutative. In particular, R º is nilpotent if it is a weakly nilpotent group or an n-Engel group for some positive integer n. We also establish that the ring R is strictly Lie-nilpotent if and only if it is an Engel ring and the quotient ring of the ring R by its Jacobson radical is commutative.
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References
V. G. Vilyatser, “On the theory of locally nilpotent groups,” Usp. Mat. Nauk, 13, No. 2, 163–168 (1958).
E. S. Golod, “Some problems of Burnside type,” in: Proceedings of the International Mathematical Congress, Mir, Moscow (1966), pp. 284–289.
G. Groza, “Artinian rings having a nilpotent group of units,” J. Algebra, 121, No. 2, 253–262 (1989).
R. Yu. Evstaf’ev, “Artinian rings with nilpotent adjoint group,” Ukr. Mat. Zh., 58, No. 3, 417–426 (2006).
Yu. Ishchuk, “On associated groups of rings,” London Math. Soc. Lect. Notes, 304, 284–293 (2003).
M. I. Khuzurbazar, “Multiplicative group of a division ring,” Dokl. Akad. Nauk SSSR, 131, No. 6, 1268–1271 (1960).
D. M. Riley and M. C. Wilson, “Associative ring satisfying the Engel condition,” Proc. Amer. Math. Soc., 127, No. 4, 973–976 (1999).
N. Gupta and F. Levin, “On the Lie ideals of a ring,” J. Algebra, 81, No. 1, 225–231 (1983).
K. W. Gruenberg, “Two theorems on Engel groups,” Proc. Cambridge Phil. Soc., 49, 377–380 (1953).
B. Amberg and Ya. Sysak, “Semilocal rings with n-Engel multiplicative group,” Arch. Math., 83, 416–421 (2004).
B. Amberg and Ya. Sysak, “Radical rings with Engel conditions,” J. Algebra, 231, 364–373 (2000).
L. Fuchs, Infinite Abelian Groups, Academic Press, New York (1970).
F. Kasch, Moduln und Ringe, Teubner, Stuttgart (1977).
J. Bjork, “Conditions which imply that subgroups of Artinian rings are Artinian,” J. Reine Math., 247, 123–138 (1971).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1264–1270, September, 2006.
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Evstaf’ev, R.Y. On Artinian rings satisfying the Engel condition. Ukr Math J 58, 1433–1440 (2006). https://doi.org/10.1007/s11253-006-0142-1
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DOI: https://doi.org/10.1007/s11253-006-0142-1