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Conditions for nilpotency of Lie rings. II. (English) Zbl 0291.17007
The author presents a correction to his paper [ibid. 79, 289–296 (1962; Zbl. 286.17008)], as well as some new results. If we define
$D_p(L)=\{x\mid x\in L,\;p^\mu x= 0 \quad\text{for some } \mu\},$
then Lemma 4.5 [loc. cit.] can now be proved by a modification of the original methods. The author establishes the following two results.
Theorem 1: Let $$L$$ be a finite dimensional Lie algebra over the Noetherian ring $$R$$. If every maximal subalgebra of $$L$$ is an ideal, then $$L$$ is $$\omega$$-nilpotent.
Theorem 2: Let $$A$$ be a torsion-free associative algebra over the integral domain $$R$$. If $$A$$ is generated as an $$R$$-module by a finite set of nilpotent elements, then $$A$$ is nilpotent.
As the author points out, Theorem 2 generalizes the well-known theorem of Wedderburn, which states that if a finite-dimensional associative algebra $$A$$ over a field has a basis of nilpotent elements, then $$A$$ is nilpotent. The author derives some consequences from Theorem 2, one of the most interesting being the following.
Theorem: If $$A$$ is an associative algebra over the Noetherian ring $$R$$ and if $$A$$ can be generated as an $$A$$-module by a finite set of nilpotent elements, then $$A$$ is $$\omega$$-nilpotent (i.e., \bigcap_{i=1}^\infty A^i=(0))$$.$$
Show Scanned Page ##### MSC:
 17B30 Solvable, nilpotent (super)algebras
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##### References:
  Barnes, D. W.: Nipotency of Lie algebras. Math. Z.79, 237-238 (1962). · Zbl 0122.04001  ?: Conditions for nilpotency of Lie rings. Math. Z.79, 289-296 (1962). · Zbl 0286.17008  Gruenberg, K. W.: The upper central series in soluble groups. Illinois J. of Math.5, 436-466 (1961). · Zbl 0244.20028  Wedderburn, J. H. M.: Note on algebras. Annals of Math. (2)38, 854-856 (1937). · Zbl 0018.10301  Zariski, O., andP. Samuel: Commutative algebra. Princeton-Toronto-New York-London 1958. · Zbl 0081.26501
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