Let $R$ be a graded ring, with the grading induced by an abelian monoid $M$. I know that if $R$ is a finitely generated $R_0$-algebra, then $Supp(R)=\{w\in W\mid R_w\neq \emptyset\}$ is a finitely generated monoid: given a set of generators it suffices to consider the graded pieces in which they belong, and these graded pieces will be the generators of $Supp(R)$.
I was wondering if also the converse is true, that is if $Supp(R)$ is finitely generated (as a monoid), can we conclude $R$ is finitely generated (as an algebra)? My gut feeling is this is not true: this is because even if some $w_1,\ldots,w_n$ generate $W$, we have no control if there exist $r_1,\ldots,r_n$, with $r_i\in R_{w_i}$, generating $R$. however I'm not very good at finding counterexamples and that's why I post it here (I'm an aspiring algebraic geometer, I came across these while studying the Cox ring and its finitely generation for Mori Dream Spaces). Thanks in advance
It's not true: take $R=k[X_1,X_2,\dots]$, the ring of polynomials in countably many indeterminates over $k$, and give it the grading by total degree. Then the grading monoid is $\mathbb{N}$, so obviously it is finitely generated, but $R$ is not finitely generated as a $R_0$-algebra (as $R_0=k$).