Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1.

In a ring $A$, let $\Sigma$ be the set of all ideals in which every element is a zero-divisor. Show that set $\Sigma$ has maximal elements and that every maximal element of $\Sigma$ is a prime ideal. Hence the set of zero-divisors in $A$ is a union of prime ideals.

I see by an application of Zorn's Lemma that $\Sigma$ has maximal elements. I take $\mathfrak{m}$ to be maximal in $\Sigma$, with $xy\in\mathfrak{m}$. Since $xy$ is a zero divisor, $xyz=0$ for some $z\neq 0$. If $yz=0$, then $y$ is a zero divisor, otherwise $x$ is a zero divisor. So I guess I then want to show $x\in\Sigma$ or $y\in\Sigma$. If neither is, then $\mathfrak{m}$ is properly contained in both $(\mathfrak{m},x)$ and $(\mathfrak{m},y)$. However, I'm not sure how to show either of these ideals is again in $\Sigma$.

If my understanding is correct, elements of $(\mathfrak{m},x)$ are finite sums of the form $\sum_ia_im_i+bx$ for $a_i\in A$, $m_i\in\mathfrak{m}$ and $b\in A$. To show this sum is a zero divisor, my hunch is that if $c_im_i=0$ for $c_i\neq 0$ and $dx=0$ for $d\neq 0$, then $$ \left(\sum_ia_im_i+bx\right)(d\prod_ic_i)=0. $$ My concern is that perhaps $d\prod_ic_i=0$, so the above wouldn't show that $(\mathfrak{m},x)$ consists of only zero divisors. How can I get around this? Or is there perhaps a better approach? Thank you for your help.

Solution 1:

Let $\frak{m}$ be a maximal element in $\Sigma$. We want to show it is prime, i.e. that if $x\notin\frak{m}$ and $y\notin\frak{m}$, then $xy\notin\frak{m}$.

If $x\notin\frak{m}$ and $y\notin\frak{m}$, then ${\frak{m}}+(x)$ and ${\frak{m}}+(y)$ are both ideals of $A$ that strictly contain $\frak{m}$, and therefore each must contain non-zero-divisors ($\frak{m}$ is maximal among ideals consisting only of zero-divisors, so any ideal strictly containing $\frak{m}$ cannot consist only of zero-divisors). Thus the ideal $({\frak{m}}+(x))({\frak{m}}+(y))\subseteq{\frak{m}}+(xy)$ contains non-zero-divisors (because there is at least one non-zero-divisor in each of ${\frak{m}}+(x)$ and ${\frak{m}}+(y)$, and the product of two non-zero-divisors is a non-zero-divisor). But the fact that ${\frak{m}}+(xy)$ contains non-zero-divisors implies that ${\frak{m}}+(xy)$ strictly contains $\frak{m}$, hence $xy\notin\frak{m}$. Thus $\frak{m}$ is prime.

Solution 2:

Hint $ $ The non-zero-divisors form a saturated monoid $\rm\:M\:$ (i.e. $\rm\:ab\in M\!\iff\! a\in M\:$ & $\rm\:b\in M$) so its complement is a union of prime ideals (this can be proved either by localization, or by a direct elementary proof using that an ideal maximal wrt the exclusion of a monoid is prime). For a nice exposition see the first few pages of Kaplansky: Commutative Rings.

Note $ $ For generalizations, see Lam and Reyes: Oka and Ako Ideal Families in Commutative Rings mentioned by Zev, and the paper reviewed below.

MR 95i:13023 13G05 06F20
Anderson, D. D.; Zafrullah, Muhammad
On a theorem of Kaplansky.
Boll. Un. Mat. Ital. A (7) 8 (1994), no. 3, 397--402.

The complement of a saturated multiplicatively closed set is a union of prime ideals (Bourbaki). $\ $ The authors apply this fact to a property (*) of non-zero elements in an integral domain such that the elements satisfying (*) form a saturated multiplicatively closed set. Suitable choices of (*) yield characterizations of UFDs [I. Kaplansky, Commutative rings, Univ. Chicago Press, Chicago, IL, 1974; MR 49 #10674], GCD domains, valuation and Prüfer domains. Lattice ordered groups and Riesz groups are characterized similarly. [Reviewed by C. P. L. Rhodes]

Zbl 816.13001

Analyzing the proof of Kaplansky's theorem, that an integral domain is a unique factorization domain if and only if every nonzero prime ideal contains a nonzero principal prime ideal, the authors state - leaving the proof to the reader:

Let D be an integral domain. Let (*) be a property of elements in D. Assume that the set S of nonzero elements in D with (*) is a nonempty saturated multiplicatively closed set. Then every nonzero element in D has (*) if and only if every nonzero prime ideal contains a nonzero element with (*). This observation is then applied to various situations, to characterize

1. integral domains that are UFD's,
2. integral domains that are valuation domains,
3. integral domains that are Pruefer domains,
4. directed partially ordered groups that are lattice-ordered,
5. directed partially ordered groups that are Riesz groups.

[ K.Roggenkamp (Stuttgart)]

Solution 3:

The following steps lead to a solution:

(1) If $a\in\Sigma$ is not a prime ideal, we will show that $a$ is not a maximal element of $\Sigma$. Since $a$ is not a prime ideal, there exists $x,y\not\in a$ such that $xy\in a$. The ideal $(a:x)=\{z\in A:xz\in a\}$ can be considered.

(2) Prove that $a\subseteq (a:x)$ and that this inclusion is proper.

(3) Prove that $(a:x)$ consists entirely of zero-divisors. (Hint: If not, then there exists $z\in (a:x)$ such that $z$ is not a zero divisor. Deduce that $(a:z)\in \Sigma$.)

(4) Show that $a$ is not maximal. (Hint: The inclusion $a\subseteq (a:z)$ is proper. Why?)

Solution 4:

An elements of $(\mathfrak{m},x)$ can be described as $m_1+a_1 x$ where $m_1 \in \mathfrak{m}$ and $a_1 \in A$, a little simpler than yours. And I think it is better to assume that $x \notin \mathfrak{m}, y \notin \mathfrak{m} $ then deduce a contradiction, since $(\mathfrak{m},x)$ alone may properly contain $\mathfrak{m}$, but both $(\mathfrak{m},x) \supsetneq \mathfrak{m}$, $(\mathfrak{m},y) \supsetneq \mathfrak{m}$ makes a contradiction.

So assume that $xy \in \mathfrak{m}$ with $(\mathfrak{m},x) \supsetneq \mathfrak{m}$, $(\mathfrak{m},y) \supsetneq \mathfrak{m}$, then there are $m_1+a_1 x \in (\mathfrak{m},x)$ and $m_2+a_2 y \in (\mathfrak{m},y)$ which are not 0-divisors. Then $(m_1+a_1 x)(m_2+a_2 y) \in \mathfrak{m}$ since $xy$ is in $\mathfrak{m}$. But this is a contradiction, since a product of non 0-divisors is not a 0-divisor.

Solution 5:

Zev's answer is excellent. However, here are some exercises that provide practice with the technique Zev introduced (all rings are commutative and have a multiplicative identity):

Exercise 1: Let $A$ be a ring with at least one non-principal ideal. Let $\Sigma$ be the set of all ideals of $A$ that are not principal. Prove that $\Sigma$ has maximal elements and that maximal elements of $\Sigma$ are prime. (Hint: study my answer carefully.)

Exercise 2: Let $A$ be a ring with at least one non-finitely generated ideal. Let $\Sigma$ be the set of all ideals of $A$ that are not finitely generated. Prove that $\Sigma$ has maximal elements and that maximal elements of $\Sigma$ are prime. (Hint: let $I$ be a maximal element of $\Sigma$. If $I$ is not prime, then there exists $x,y\not\in I$ such that $xy\in I$. Use the fact that $I+(x)$ is finitely generated and that $(I:x)$ is finitely generated. Why are these ideals finitely generated?)

Exercise 3: Let $S$ be a multiplicatively closed subset of $A$ such that $0\not\in S$. (Definition: $S$ is closed under multiplication and $1\in S$.) Let $\Sigma$ be the set of all ideals $I$ such that $I\cap S = \emptyset$. Note that $\Sigma\neq \emptyset$ since $0\in \Sigma$. Prove that $\Sigma$ has maximal elements and that a maximal element $I$ of $\Sigma$ is a prime ideal. (Hint: use the technique Zev introduced.)