singularities of $R^G$ is necessary. When $R^G$ has isolated singularities, the representations of the singularities are well understood (cf. [@MU1; @MU2; @U]). Iyama-Wemyss developed a wonderful theory on the representations of non-isolated singularities when the algebra under consideration is commutative (cf. [@IW]). The representations of noncommutative non-isolated singularities remain less understood. The main purpose of this paper is to understand the properties of the invariant subalgebra $R^G$ when it has non-isolated singularities.

The main tool we will use in this paper is the [*radical*]{} $\mfr(R,G)$ of the group action. Let $B=R*G$ be the skew group ring, and let $e=\frac{1}{|G|}\sum_{g\in G}g\in \kk G$, the integral of $\kk G$. The radical $\mfr(R,G)$ is in fact the the intersection $R\cap BeB$. The radical can be introduced even if the group $G$ is not finite. We introduce the concept of two [*pertinent sequences* ]{} of elements of $R$ (cf. Section \[sec1\]), through which we define the radical $\mfr(R,G)$ for any group action. The radical $\mfr(R,G)$ is an ideal of $R$. We show in Section \[sec2\] that $\mfr(R,G)$ fits into the the classical radical theory which justifies the name. It is usually difficult to compute $R\cap BeB$ directly, while there are several natural ways to 