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Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

Research paper by N.Yu. Makarenko

Indexed on: 27 Apr '18Published on: 15 Apr '18Published in: Journal of Algebra



Abstract

Publication date: 15 July 2018 Source:Journal of Algebra, Volume 506 Author(s): N.Yu. Makarenko Let A be an associative algebra of arbitrary dimension over a field F and G a finite group of automorphisms of A of order n, prime to the characteristic of F. Denote by A G = { a ∈ A a g = a for  all g ∈ G } the fixed-point subalgebra. By the classical Bergman–Isaacs theorem, if A G is nilpotent of index d, i.e. ( A G ) d = 0 , then A is also nilpotent and its nilpotency index is bounded by a function depending only on n and d. We prove, under the additional assumption of solubility of G, that if A G contains a two-sided nilpotent ideal I ◁ A G of nilpotency index d and of finite codimension m in A G , then A contains a nilpotent two-sided ideal H ◁ A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d. An even stronger result is provided for graded associative algebras: if G is a finite (not necessarily soluble) group of order n and A = ⨁ g ∈ G A g is a G-graded associative algebra over a field F, i.e. A g A h ⊂ A g h , such that the identity component A e has a two-sided nilpotent ideal I e ◁ A G of nilpotency index d and of finite codimension m in A e , then A has a homogeneous nilpotent two-sided ideal H ◁ A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of n, d and m.