Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F.
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References
G. Letzter, “Derivations and nil ideals,” Rend. Circolo Mat. Palermo, 37, No. 2, 174–176 (1988).
B. Hartley, “Locally nilpotent ideals of a Lie algebra,” Proc. Cambridge Philos. Soc., 63, 257–272 (1967).
C. Lanski, “Left ideals and derivations in semiprime rings,” J. Algebra, 277, Issue 2, 658–667 (2004).
D. A. Jordan, “Noetherian Ore extensions and Jacobson rings,” J. London Math. Soc., 10, No. 2, 281–291 (1975).
V. S. Luchko and A. P. Petravchuk, “On one-sided ideals of associative rings,” Alg. Discrete Math., No. 3, 1–6 (2007).
N. Jacobson, Lie Algebras, Interscience, New York (1962).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 7, pp. 1000–1004, July, 2009.
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Luchko, V.S. On the action of derivations on nilpotent ideals of associative algebras. Ukr Math J 61, 1187–1191 (2009). https://doi.org/10.1007/s11253-009-0259-0
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DOI: https://doi.org/10.1007/s11253-009-0259-0