Why we need $R$ to be radical at here?

Question

Jacob Lurie used one example for the theorem of Scholism:

Let $R=k[X]$ be the coordinate ring of a variety $X$ in $\mathbb{C}^{n}$. Assume $X$ is reduced. Then $MaxSpecR$ is a union of irreducible components $X_{i}$, which are the closures of the minimal primes of $R$. The fields you get by localizing at the minimal primes depend only on the irreducible components, and in fact are the rings of meromorphic functions on $X_{i}$. Indeed we have a map $$k[X]\rightarrow \prod k[X_{i}]\rightarrow \prod k(X_{i})$$

If we do not assume $R$ is radical, this is not true.

I am really confused with what he wrote. I have some really elementary questions to ask:

1) Why $R$ is radical matters at here?

2) Why $MaxSpec(R)$ is the union of irreducible components $X_{i}$? If I am not mistaken I think he means the maximum spectrum at here. I understand that points in $X$ corresponds to elements in $MaxSpec(R)$, however why each irreducible component corresponds to a point in $MaxSpec(R)$?

3) How should I understand "..in fact are rings of meromorphic functions on $X_{i}$?" I feel very confused.

@Georges: From Akhil Mathew's notes (21.11), p. 92, it looks like a scholium met a porism. I suspect they indulged in holism and, unexpectedly, there resulted a scholism. We should proactively engage in action against such abus de langage... — Martin, Jan 24 '13 at 21:27
The scholism in question is stated with an irrelevant Noetherian hypothesis and I think a more canonical proof that a reduced ring is a subring of a product of fields is to naturally embed it into the product $\prod_{\mathfrak p\in Spec(R)} \kappa( R/\mathfrak p)$ for all primes $\mathfrak p \subset R$ of the residue fields $\kappa( \mathfrak p)=Frac(R/\mathfrak p)$ . — Georges Elencwajg, Jan 24 '13 at 22:14
Sorry for not providing the background. Yes, it is from Matthew's notes. — Bombyx mori, Jan 24 '13 at 22:22
@Martin: I emailed Lurie as politely as possible, but did not receive back anything yet. I will moving on with other stuff if he did not write me back. — Bombyx mori, Jan 25 '13 at 09:00
I think the right person to mail to would be Akhil (see the introduction of the notes). — Martin, Jan 25 '13 at 09:09
@Martin: So far did not receive anything from Lurie. I am forwarding this to Akhil. — Bombyx mori, Jan 25 '13 at 18:57

Why we need $R$ to be radical at here?

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