Ring of rational functions for reducible variety

Question

Let $X$ be some affine algebraic variety over $\mathbb{k}$ (i.e. some closed subset in $\mathbb{A}_\mathbb{k}^n$). First suppose $X$ to be irreducible. Then the algebra $\mathbb{k}[X]$ is a domain and we can consider the field of rational functions $\mathrm{Quot}_{\mathbb{k}[X]}=\mathbb{k}(X)$. Could you explain me how to build an analogue of this field in the case when $X$ is not necessarily irreducible? Then $\mathbb{k}[X]$ must not be a domain and we are to build some kind of localization?
Also, what is the destination of rational functions? Why we cannot be satisfied with only regular maps and regular functions?

Answer 1

The analogue of the quotient field for a ring with zero divisors is the total ring of fractions: basically, just invert everything that is not a zero divisor. Geometrically, an element of this ring can be viewed as a collection of rational functions, one on each irreducible component of $X$, such that they coincide on intersections.

Rational functions are important for a wide variety of reasons. Asking this question is like asking why $\mathbf Q$ is important. Why weren't we happy with $\mathbf Z$? Well, it wasn't big enough for what we wanted to do, so we enlarged it.

Answer 2

This question may have general interest. There is an exercise in Matsumura (exercise 9.11 page 70) saying the following:

Let $A$ be a noetherian ring with the propery that $A_{\mathfrak{p}}$ is a domain for all prime ideals $\mathfrak{p}$. Let $\mathfrak{p}_1,..,\mathfrak{p}_l$ be the minimal prime ideals of $A$. It follows there is an isomorphism of rings

I1. $p:A \cong A/\mathfrak{p}_1\oplus \cdots \oplus A/\mathfrak{p}_l:=A_1\oplus \cdots \oplus A_l$

defined by

$p(a):=(\overline{a}, ..., \overline{a})$. This gives at the level of schemes an isomorphism

$Spec(A) \cong Spec(A_1) \cup \cdots \cup Spec(A_l)$

where the union is the "disjoint union".

This gives a general formula for the total ring of fractions $Tot(A)$:

I2. $Tot(A)\cong K(A/\mathfrak{p}_1)\oplus \cdots \oplus K(A/\mathfrak{p}_l)$

where $K(A/\mathfrak{p}_i)$ is the quotient field of the integral domain $A/\mathfrak{p}_i$.

Question: "Then k[X] must not be a domain and we are to build some kind of localization?"

Answer: The "total ring of fractions" is defined on page 21 in Matsumura's book "Commutative Ring Theory". This is a construction generalizing the "quotient field" defined for integral domains. If the ring $k[X]$ satisfies formula I1, it follows I2 gives a general formula for $Tot(k[X])$.

PS: I have not done the exercise myself.