Note that for all $i\neq j$, being $\ell_{i:j}=p_j$, elements in are given by $$\xi_{i,j}=\zeta_i^*\zeta_j \, .$$ Named $p:=\prod_{j=0}^np_j$, since $\alpha_t(\zeta_i)=t^p\zeta_i$ for all $i=0,\ldots,n$ ($p_i\ell_i=p$ for all $i$), an invariant monomial $\zeta^{{\underline}{k}}$ contains the same number of $\zeta_i$’s and of $\zeta_i^*$’s (each counted with multiplicities). Using the relation the factors can be reordered so that $\zeta_i$’s and $\zeta_i^*$’s are alternating, i.e. we can write an invariant $\zeta^{{\underline}{k}}$ as a product of elements $\xi_{i,j}=\zeta_i^*\zeta_j$.

For the classes of spaces in Theorem \[thm:11\], it is clear from the proof that ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ is a of the algebra generated by the elements: $$\label{eq:genLq}
x_i:=z_iz_i^* \;,\quad
\zeta_i:=z_i^{p_i} \;,\quad
\zeta_i^* \;, \qquad \textup{for} \quad i=0,\ldots,n \, .$$ If ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ with ${\underline}{p}$ pairwise coprime, this algebra is the lens space algebra ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ defined in , for the action of the cyclic group ${\mathbb{Z}}_p$, with parameter $p$ now given by $p:=p_0p_1\ldots p_n$.

\[thm:13\] Let ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ with ${\underline}{p}$ pairwise coprime, and let $p:=p_0p_1\ldots p_n$. Then the algebra ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ is generated by the elements .

By arguing as in the proof of Lemma \[lemma:2\], clearly ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ is generated by the elements $x_i=z_iz_i^*$ and by monomials $z^{{\underline}{k}}$ that are ${\mathbb{Z}}_p$-invariant, which happens if and only if ${\underline}{k}\cdot{\underline}{\ell}\in p{\mathbb{Z}}$. Using ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ the invariance 