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Let A be an associative algebra of arbitrary dimension over a field F and G afinite soluble group of automorphisms of A oforder n, prime to thecharacteristic of F. We prove that if the fixed-point subalgebra of A under theaction of G contains a two-sided nilpotent ideal of nilpotency index d and offinite codimension m in I, then A contains a nilpotent two-sided ideal ofnilpotency index bounded by a function of n and d and of finite codimensionbounded by a function of m, n and d.

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