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Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of $R$-modules $\varphi: S \to R$ that preserves degrees and takes each element of $R$ to itself, then $R$ is a finitely generated $k$-algebra.
Proof. Let $\hat{m} \subset R$ be the ideal generated by the homogeneous elements of $R$ of strictly positive degree. Since $S$ is Noetherian, the ideal $\hat{m}S$ has a finite set of generators, which may be chosen to be homogeneous elements $f_1, \dots, f_s$ of $\hat{m}$. We shall show that these elements generate $R$ as a $k$-algebra.
Why is the bolded part true?