Galois extensions of radical algebras

Suppose $G$ is a finite group of automorphisms of an associative algebra $K$ with an identity element over a field $F$. Let $t(x)=\sum_{g\in G}x^g$. Assume that $\rho$ is a supernilpotent radical which is closed under the taking of subalgebras and satisfies the following condition: if $A\in\rho$ and $M$ is a nonempty set, then the ring $A_M$ of $M\times M$ matrices all but a finite number of whose columns are zero is radical.
THEOREM. If $R$ is a two-sided ideal of $K$ and $K=t(K)K,$ then $t(R)\in\rho$ implies $R\in\rho$.
Examples of radicals satisfying the above conditions are Baer's lower radical, the locally nilpotent radical, the locally finite radical, and also the algebraic kernel and Köthe radical, if $F$ is uncountable.
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