the convenience of the narratives, we conventionally write $(a_1,\dots,a_n)\vee (a'_1,\dots,a'_m)$ for the sequence $(a_1,\dots,a_n,a'_1,\dots,a'_m)$. We have the following basic properties of pertinent sequences.

\[prop-basic\] Assume $(a_1,\dots,a_n)\overset{G}\sim(b_1,\dots,b_n)$. The following hold:

-   $(h\cdot a_1,\dots,h\cdot a_n)\overset{G}\sim(h\cdot b_1,\dots,h\cdot b_n)$ for all $h\in G$;

-   $(a a_1,\dots,aa_n)\overset{G}\sim(b_1b,\dots, b_nb)$ for all $a,b\in A$;

-   if $(a'_1,\dots,a'_m)\overset{G}\sim(b'_1,\dots,b'_m)$, then $(a_1,\dots,a_n)\vee (a'_1,\dots,a'_m)\overset{G}\sim(b_1,\dots ,b_n)\vee (b'_1,\dots,b'_m)$.

Straightforward.

Given a $G$-action $\theta:G\to \operatorname{Aut}(R)$, we define $$\mfr(R,G)=\left\{\sum_{i=1}^na_ib_i|(a_1,\dots,a_n)\overset{G}\sim(b_1,\dots,b_n),n\ge1\right\}.$$ By Proposition \[prop-basic\](ii), $\mfr(R,G)$ is closed under addition, and by Proposition \[prop-basic\](ii) $\mfr(R,G)$ is an ideal of $R$.

\[def-pideal\] We call $\mfr(R,G)$ [*the radical of $R$ under the $G$-action*]{}, or simply, [*the $G$-radical of $R$*]{}, and we call the quotient algebra $\mathfrak{R}(R,G):=R/\mfr(R,G)$ the [*pertinency algebra*]{} of $R$ under the $G$-action.

\(i) We will show in the next section that the ideal $\mfr(R,G)$ fits into the classical radical theory of rings (cf. [@Sz]).

\(ii) The $G$-radical $\mfr(R,G)$ depends mainly on the group action $\theta:G\to \operatorname{Aut}(R)$.

\(i) Let $R=\kk[x,y]$. Let $\sigma$ be the automorphism of $A$ defined by $\sigma(x)=y$ and $\sigma(y)=x$. Let $G=\{1,\sigma\}$. Note that $(x,1)$ and $(1,-y)$ are pertinent sequences. Hence $x-y\in \mfr(R,G)$. We claim that $x^n$ ($n\ge2$) is not in $\mfr(R,G)$. Indeed, if $x^n\in \mfr(R,G)$, then there are pertinent sequences $(a_1,\dots,a_m)$ and $(b_1,\dots,b_m)$ such that $x^n=\sum_{i=1}^ma_ib_i$. 