Note that $R$ has a basis $\{x^sy^t|s,t\ge0\}$. By a simple comparison of powers of $y$ in the sum $\sum_{i=1}^ma_ib_i$, we may assume that $a_i=\sum_jk_{i_j}x^{s_{i_j}}$ and $b_i=r_ix^{t_i}$, where $k_{i_j},r_i\in\kk$ and $t_1,\dots t_m$ are different numbers. Since $(a_1,\dots,a_m)$ and $(b_1,\dots,b_m)$ are pertinent sequences, we have $0=\sum_{i=1}^ma_i\sigma(b_i)=\sum_{i=1}^m\sum_jk_{i_j}r_ix^{s_{i_j}}y^{t_i}$. Then $\sum_jk_{i_j}r_ix^{s_{i_j}}=0$ for all $i$. Therefore, $\sum_{i=1}^ma_ib_i=\sum_{i=1}^m\sum_jk_{i_j}r_ix^{s_{i_j}}x^{t_i}=0$, a contradiction. Hence $x^n$ is not in $\mfr(R,G)$ for all $n\geq 1$. It follows that $\mathfrak{R}(R,G)\cong \kk[x]$.

\(ii) Let $R=\kk_{-1}[x,y]$. Let $\sigma$ be the automorphism of $R$ defined by $\sigma(x)=y$ and $\sigma(y)=x$. Let $G=\{1,\sigma\}$. As we see in Example \[example1\], the sequences $(x,y)$ and $(x,y)$ are pertinent, $x^2+y^2\in \mfr(R,G)$. Similar to (i), $x-y\in \mfr(R,G)$. If $char\kk\neq 2$, then we see $x^2,y^2,xy\in \mfr(R,G)$. Therefore $\mfr(R,G)=\kk(x-y)\oplus R_2\oplus R_3\oplus\cdots$, and $\mathfrak{R}(R,G)$ is of dimension 2.

In some cases, the $G$-radical may be trivial. For example, if the $G$-action on $R$ is trivial, that is, $g\cdot a=a$ for all $g\in G$ and $a\in R$, then one sees that the $G$-radical is zero. Following the classical radical theory, we introduce the following definition.

Let $R$ be a left $G$-module algebra. If the $G$-radial $\mfr(R,G)=0$, then we say that the $G$-action $R$ is [*$G$-semisimple*]{}.

By the definition of the $G$-radical, one sees the following basic 