Why $\cap \frak{n}^j=0$?

Question

From David Eisenbud's commutative algebra book (page $199)$.

I have some understanding problem in the proof of Theorem $7.16$ as follows:

My questions:

$(1)$ What type elements in $S$ ? Are they necessarily power series over $R$? See the highlighted line in the proof of part $(a)$, $ \ \ g \in S$ means what ?? Is $g \in S$ a power series ?

$(2)$ In the proof of part $(b)$, (see highlighted line), why $\cap \frak{n}^j=0$ ? What is $j$ here ?? Is $j <i$ here ?

Answer 1

(1) $S$ is just an arbitrary $R$ algebra that is complete with respect to $\frak n$. The highlighted line does not say that $g\in S$, but rather talks about the image of $g$ in $S$, i.e., about $\phi(g)$. Of course $g\in R[\![x_1,\ldots,x_n]\!]$ is a power series.

(2) Recall what it means to be complete with respect to $\frak n$. Then again, it would be less confusing to explicitly write $\bigcap_{\color{red}{j\in\Bbb N}}\frak n^j$.