Indeed, this is the case when $R/R^G$ is a [*Hopf Galois extension*]{} (for the definition, see [@CFM]).

\[prop3-1\] Let $G$ be a finite group and let $R$ be a left $G$-module algebra. Then $\mfr(R,G)=R$ if and only if $R/R^G$ is a Hopf Galois extension.

Let $e=\frac{1}{|G|}\sum_{g\in G} g\in \kk G$. Note that $\mfr(R,G)=R$ if and only if $ReR=R*G$, if and only if $R^G$ and $R$ are Morita equivalent, which is equivalent to the condition that $R/R^G$ is Hopf Galois extension [@CFM Theorem 1.2].

Assume that $R$ has finite GK-dimension. It is clear that if $R/R^G$ is a Hopf Galois extension, then $\Pty(R,G)=\operatorname{GKdim}(R)$. A weak version of a Hopf Galois extension, called a [*Hopf dense Galois extension*]{}, was introduced in [@HVZ]. The next result is a consequence of Lemma \[lem1\] and [@HVZ Proposition 1.3].

Assume that $G$ is a finite group. Let $R$ be a noetherian left $G$-module algebra with finite GK-dimension. Then $R/R^G$ is a Hopf dense Galois extension if and only if $\Pty(R,G)=\operatorname{GKdim}(R)$.

The following result gives an example of Hopf dense Galois extensions.

Let $R=\kk\oplus R_1\oplus R_2\oplus\cdots$ be a neotherian connected graded algebra generated by elements $x_1,\dots,x_m$ of degree one with finite GK-dimension. Let $G$ be the cyclic 