in the radical ideal.

\[lem2\] Let $R$ be an algebra, and $G$ a cyclic group generated by $\sigma$. Assume that there are elements $a_1,\dots,a_n\in R$ such that $\sigma\cdot a_i=\xi a_i$ for $i=1,\dots,n$, where $\xi$ is an $n$-th primitive root of unity. If $\operatorname{char}\kk\nmid n$, then $a_1a_2\cdots a_n\in \mfr(R,G)$.

We show that the following sequences are pertinent under the $G$-action: $(1,a_1,a_1a_2,\dots,a_1a_2\cdots a_{n-1})$ and $(a_1a_2\cdots a_n,a_2\cdots a_n,\dots,a_n)$. Note that $\sigma^j\cdot a_i=\xi^j a_i$. For $j=1,\dots,n-1$, we have $\sum_{i=0}^{n-1}a_1\cdots a_i(\sigma^j\cdot (a_{i+1}\cdots a_n))=(\sum_{i=0}^{n-1}\xi^{j(n-i)})a_1a_2\cdots a_n=0$. Hence $a_1a_2\cdots a_n\in \mfr(R,G)$.

Let $R$ be an algebra, and let $a_1,\dots,a_k\in R$. We write $a_1\cdots \hat{a}_i\cdots a_k$ for the product $a_1\cdots a_{i-1}a_{i+1}\cdots a_k$ with $a_i$ missing. Similarly, we have the notion $a_1\cdots \hat{a}_{i_1}\cdots \hat{a}_{i_2}\cdots \hat{a}_{i_s}\cdots a_k$ for $1\leq i_1<\dots<i_s\leq k$. We write $$a_{\widehat{[i_1i_2\cdots i_s]}}:=a_1\cdots \hat{a}_{i_1}\cdots \hat{a}_{i_2}\cdots \hat{a}_{i_s}\cdots a_k,$$ and $$(a_{\widehat{[i_1i_2\cdots i_s]}}:1\leq i_1<\dots<i_s\leq k)$$ for the sequence corresponding to all the possible choices of arrays $[i_1i_2\cdots i_s]$ in the lexicographic order. Similarly, we have $$a_{[i_1i_2\cdots i_s]}:=a_{i_1}\cdots a_{i_s}$$ for the products of elements, and $$(a_{[i_1i_2\cdots i_s]}:1\leq i_1<\dots<i_s\leq k).$$ Let $G$ be a group acting on $R$. For $g_1,\dots,g_k\in G$, we write [$$g_{\widehat{[i_1i_2\cdots i_s]}}(a_{\widehat{[i_1i_2\cdots i_s]}}):=g_1(a_1)\cdots \hat{a}_{i_1}\cdots \hat{a}_{i_2}\cdots \hat{a}_{i_s}\cdots g_k(a_k),$$]{} and $$\left(g_{\widehat{[i_1i_2\cdots i_s]}}(a_{\widehat{[i_1i_2\cdots i_s]}}):1\leq i_1<\dots<i_s\leq k\right)$$ for the corresponding sequences in the lexicographic order.

