Abstract
It is shown that a locally nilpotent ring with maximality condition for two-sided ideals is nilpotent. The restriction on the characteristic in one of the author's previously published theorems is lifted. A one-sided nil-ideal of an alternative ring, satisfying the maximality condition for right ideals, is a nilpotent ring. An example is constructed of a commutative locally nilpotent ring A with maximality condition for ideals which is idempotent: A = A2.
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K. A. Zhevlakov, “Notes on simple alternative rings,” Algebra i Logika,6, No. 2, 21–33 (1967).
K. A. Zhevlakov, “Nil-ideals of an alternative ring satisfying the maximality condition,” Algebra i Logika,6, No. 4, 19–26 (1967).
M. Slater, “Prime alternative rings II,” J. Algebra,15, No. 2, 244–251 (1970).
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The article was prepared for print by the author and submitted after his death which took place on February 24, 1972.
Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 121–126, August, 1972.
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Zhevlakov, K.A. A note on locally nilpotent rings with chain conditions. Mathematical Notes of the Academy of Sciences of the USSR 12, 507–509 (1972). https://doi.org/10.1007/BF01095006
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DOI: https://doi.org/10.1007/BF01095006