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

JOURNAL OF ALGEBRA

卷:506

Article

Makarenko, N. Yu.¹ 

[1] Sobolev Inst Math, Novosibirsk 630090, Russia

关键词: Associative algebra;    Actions of finite groups;    Finite grading;    Graded associative algebra;    Fixed-point subalgebra;    Almost nilpotency;   

DOI  :  10.1016/j.jalgebra.2018.03.019

来源: Elsevier

【 摘 要 】

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 is an element of A vertical bar alpha(9) = a for all g is an element of 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 1 (sic) A(G) of nilpotency index d and of finite codimension m in A(G), then A contains a nilpotent two-sided ideal H (sic) A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of ns, 1z ayd 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 = circle plus(g is an element of G) A(g) is a G-graded associative algebra over a field F, i.e. A(g)A(h) subset of A(gh), such that the identity component Ae has a two-sided nilpotent ideal I-e (sic) A(G) of nilpotency index d and of finite codimension m in A(e), then A has a homogeneous nilpotent two-sided ideal H (sic) 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. (C) 2018 Elsevier Inc. All rights reserved.

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