Let $k$ be a field and $S=k[x_1,\dots,x_n]$ the polynomial ring with the usual grading. Let $M$ be a finitely generated graded $S$-module.
Question 1: How can we see that $\dim M = 0$ implies that the length of $M$ is finite, i.e. $l(M)< \infty$?
Remark 1: If $R$ was local Noetherian, then the result would follow immediately from the Fundamental Theorem of Dimension Theory. I suspect that the analogue of this theorem holds for *local rings as well, even though i can't find a source that verifies that.
Question 2: Does the fundamental theorem of dimension theory for local Noetherian rings admit an analogue for *local rings? Any reference in the literature?
Remark 2: By Remark 1 we have that $\dim M_P =0$ for every $P \in \operatorname{Supp}M$ and so $l(M_P) < \infty$. This motivates
Question 3: Is it true that for a finitely generated module $M$ over a Noetherian ring $R$ (could also be local or *local) we have that $l(M) = \sup \left\{(l(M_P) : P \in \operatorname{Supp} M \right\}$?