I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here.

Let $T=\bigoplus_{\underline n\in \mathbb Z^r}T_{\underline n}$ be an $r$-graded ring defined over a local ring. Let $S=\bigoplus_{n_j=0}T_{\underline n}$ and $\mathfrak M$ be maximal homogeneous ideal of $S.$

I do not understand why $T_{\mathfrak M}$ can be considered as $\mathbb Z$-graded ring defined over the local ring $S_{\mathfrak M}.$

If I consider $S=\bigoplus_{n_i,n_j=0}T_{\underline n}$ and $\mathfrak M$ is maximal homogeneous ideal of $S,$ then also the statement is true?

Please explain the change of grading used in the proof.