Section 10.25 (02LV): Zerodivisors and total rings of fractions

10.25 Zerodivisors and total rings of fractions

The local ring at a minimal prime has the following properties.

Lemma 10.25.1. Let $\mathfrak p$ be a minimal prime of a ring $R$ . Every element of the maximal ideal of $R_{\mathfrak p}$ is nilpotent. If $R$ is reduced then $R_{\mathfrak p}$ is a field.

Proof. If some element $x$ of ${\mathfrak p}R_{\mathfrak p}$ is not nilpotent, then $D(x) \not= \emptyset$ , see Lemma 10.17.2. This contradicts the minimality of $\mathfrak p$ . If $R$ is reduced, then ${\mathfrak p}R_{\mathfrak p} = 0$ and hence it is a field. $\square$

Lemma 10.25.2. Let $R$ be a reduced ring. Then

$R$ is a subring of a product of fields,
$R \to \prod _{\mathfrak p\text{ minimal}} R_{\mathfrak p}$ is an embedding into a product of fields,
$\bigcup _{\mathfrak p\text{ minimal}} \mathfrak p$ is the set of zerodivisors of $R$ .

Proof. By Lemma 10.25.1 each of the rings $R_\mathfrak p$ is a field. In particular, the kernel of the ring map $R \to R_\mathfrak p$ is $\mathfrak p$ . By Lemma 10.17.2 we have $\bigcap _{\mathfrak p} \mathfrak p = (0)$ . Hence (2) and (1) are true. If $x y = 0$ and $y \not= 0$ , then $y \not\in \mathfrak p$ for some minimal prime $\mathfrak p$ . Hence $x \in \mathfrak p$ . Thus every zerodivisor of $R$ is contained in $\bigcup _{\mathfrak p\text{ minimal}} \mathfrak p$ . Conversely, suppose that $x \in \mathfrak p$ for some minimal prime $\mathfrak p$ . Then $x$ maps to zero in $R_\mathfrak p$ , hence there exists $y \in R$ , $y \not\in \mathfrak p$ such that $xy = 0$ . In other words, $x$ is a zerodivisor. This finishes the proof of (3) and the lemma. $\square$

The total ring of fractions $Q(R)$ of a ring $R$ was introduced in Example 10.9.8.

Lemma 10.25.3. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset consisting of nonzerodivisors. Then $Q(R) \cong Q(S^{-1}R)$ . In particular $Q(R) \cong Q(Q(R))$ .

Proof. If $x \in S^{-1}R$ is a nonzerodivisor, and $x = r/f$ for some $r \in R$ , $f \in S$ , then $r$ is a nonzerodivisor in $R$ . Whence the lemma. $\square$

We can apply glueing results to prove something about total rings of fractions $Q(R)$ which we introduced in Example 10.9.8.

Lemma 10.25.4. Let $R$ be a ring. Assume that $R$ has finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ t$ , and that $\mathfrak q_1 \cup \ldots \cup \mathfrak q_ t$ is the set of zerodivisors of $R$ . Then the total ring of fractions $Q(R)$ is equal to $R_{\mathfrak q_1} \times \ldots \times R_{\mathfrak q_ t}$ .

Proof. There are natural maps $Q(R) \to R_{\mathfrak q_ i}$ since any nonzerodivisor is contained in $R \setminus \mathfrak q_ i$ . Hence a natural map $Q(R) \to R_{\mathfrak q_1} \times \ldots \times R_{\mathfrak q_ t}$ . For any nonminimal prime $\mathfrak p \subset R$ we see that $\mathfrak p \not\subset \mathfrak q_1 \cup \ldots \cup \mathfrak q_ t$ by Lemma 10.15.2. Hence $\mathop{\mathrm{Spec}}(Q(R)) = \{ \mathfrak q_1, \ldots , \mathfrak q_ t\}$ (as subsets of $\mathop{\mathrm{Spec}}(R)$ , see Lemma 10.17.5). Therefore $\mathop{\mathrm{Spec}}(Q(R))$ is a finite discrete set and it follows that $Q(R) = A_1 \times \ldots \times A_ t$ with $\mathop{\mathrm{Spec}}(A_ i) = \{ \mathfrak {q}_ i\}$ , see Lemma 10.24.3. Moreover $A_ i$ is a local ring, which is a localization of $R$ . Hence $A_ i \cong R_{\mathfrak q_ i}$ . $\square$

Comments (4)

Comment #6046 by Shurui Liu on April 09, 2021 at 22:51

Lemma 00EW (3) doesn't need R to be reduced and hence lemma 02LX needn't require $\mathfrak q_1 \cup \ldots \cup \mathfrak q_ t$ to be the set of zerodivisors of R (since it is automatically true).

Comment #6047 by Johan on April 10, 2021 at 12:59

@#6046. No. I suggest trying to make a counterexample.

Comment #8084 by Miles Reid on January 24, 2023 at 07:43

https://stacks.math.columbia.edu/tag/02LV

I find your Lemma 02LX fishy. Suppose R = k[x,y]/(x*y). The zero divisors of R are the two minimal prime ideals (x) and (y). The multiplicative set S consists of anything not in (x) or (y) -- so anything with a constant term, or anything like x^n + y^m. The total ring of fractions is certainly a subring of the Cartesian product of two fields k(x) x k(y). I believe that it consists of pairs f(x), g(y) such that both are regular functions at x=0 and y=0, and take the same value f(0)=g(0). In particular the elements (1,0) and (0,1) of the Cartesian product are not in S^-1R.

Matsumura's 2 books and the Wikipedia page on total ring of fractions just say the obvious definition without discussing what it means or any serious examples. In your treatment, I think Q(R) should be a subring of the Cartesian product

A slightly more complicated example would be k[x,y]/(x^n,y^m), where the total ring of fractions is a subring of k(x)[y]/y^m x k(y)[x]/x^n with the first and second factors equal modulo smaller powers of x,y. Tricky to state correctly, although the idea is clear.

I'm trying to read Mumford, Curves on a surface, Lecture 9: Cartier divisors.

Best, Miles

Comment #8090 by Stacks Project on January 26, 2023 at 17:10

Dear Miles, what about $(x + y)/(x + 2y)$ ? This corresponds to the function which is $1$ when $y = 0$ and $2$ when $x = 0$ , so it corresponds to $(1, 2)$ in $k(x) \times k(y)$ . OK?

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10.25 Zerodivisors and total rings of fractions

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