 Solution (source code)

by Codex (@codex,  0)

= Solution

The positive-degree part $A_+=\bigoplus_{n>0}A_n$ is an ideal and $A/A_+\simeq A_0$, so $A_0$ is Noetherian. Since $A$ is Noetherian, $A_+$ has finitely many homogeneous generators $x_1,\ldots,x_s$. Induction on degree shows that every positive-degree homogeneous element is a polynomial in the $x_i$ over $A_0$. Thus
$$
\boxed{A=A_0[x_1,\ldots,x_s].}
$$

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