$R/I$ is $G$-semisimple, we have $\sum_{i=1}^n\overline{a_i}\overline{b_i}=0$ in $R/I$. Therefore, $\sum_{i=1}^na_i b_i\in I$, and hence $\mfr(R,G)\subseteq I$.

We have the following “universal property” of the algebra morphisms which respect the group actions.

\[prop-univ\] Let $G$ be a finite group, and let $R$ be a left $G$-module algebra. Let $S$ be a left $G$-semisimple $G$-module algebra. For any $G$-module algebra morphism $f:R\to S$, there is a unique $G$-module algebra morphism $g:\mathfrak{R}(R,G)\to S$ such that the following diagram commutes $$\xymatrix{
  R \ar[d]_{\pi} \ar[r]^{f} &     S,   \\
  \mathfrak{R}(R,G) \ar[ur]_{g}                     }$$ where $\pi$ is the natural projection.

Since $S$ is $G$-semisimple and $f$ is also a $G$-module morphism, the subalgebra im$(f)$ of $S$ is also $G$-semisimple by Proposition \[prop-subalg\]. Hence the $G$-radical $\mfr(R,G)$ is contained in $\ker(f)$ by Lemma \[lem-min\]. Thus the assertion follows.

Let $G$ be a finite group, $R$ a left $G$-module algebra, and $B=R*G$ the skew polynomial algebra. Let $\iota: R\to B, a\mapsto a\otimes 1$ be the embedding map, so that $R$ is viewed as a subalgebra of $B$. Similarly, we view $\kk G$ as a subalgebra of $B$. Set $e=\frac{1}{|G|}\sum_{g\in G}g\in \kk G$. Then $e$ is an idempotent in $B$. Let $\mathcal{M}_R$ be the class of right $R$-modules obtained from right 