Can a product of non-principal ideals be principal, in a local ring?

Question

In a commutative ring $R$, we can define the ideal class monoid $\mathrm{ICl}(R)$ of $R$ as follows. The elements of $\mathrm{ICl}(R)$ are equivalence classes of invertible ideals (where an ideal $I$ is invertible if there exists an ideal $J$ with $IJ$ being principal). Two ideals $I$ and $J$ are equivalent if there exist nonzero $a,b \in R$ with $aI=bJ$. The resulting quotient is a (commutative) monoid under ideal product, with identity the equivalence class of any principal ideal.[1]

Q: Suppose we have two ideals $I$, $J$ in a local ring $R$, with their product $IJ = (a)$ being principal, and $a$ is not a zerodivisor.[2] Then are $I$ and $J$ themselves principal?

In particular: Is the ideal class monoid of a local integral domain always trivial?

This is probably answerable with only elementary commutative algebra, but I couldn't see it. A few related examples that motivated the question are:

• The class group $\mathrm{Cl}(R)$ of a Dedekind domain $R$. In a Dedekind domain, all ideals are invertible so $\mathrm{ICl}(R)$ recovers the usual definition of $\mathrm{Cl}(R)$. This group is often nontrivial, but for local Dedekind domains (i.e. discrete valuation rings), it is always trivial: it is easy to see that all ideals in a discrete valuation ring are principal.
• The (Weil) divisor class group of a scheme. For simplicity, consider the case of $\mathrm{Spec}(R)$, with $R$ an integrally closed Noetherian domain. We can then define the Weil class group $\mathrm{Cl}(R)$ as the group of Weil divisors modulo principal divisors, i.e. $\mathbb{Z}$-linear combinations of height $1$ prime ideals of $R$ modulo principal divisors. This also can be nontrivial, even when $R$ is local. The typical example is $R = \mathbb{C}[[x,y,z]]/(xy-z^2)$, where the divisor given by $I = (x,z)$ is non-principal, yet $2 I$ is principal, being the divisor of the rational function $x$.

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I would appreciate any geometrical intuition towards $\mathrm{ICl}(R)$, for any $R$. Does it relate in any meaningful way to the geometry of $\mathrm{Spec}(R)$, e.g. to its Weil divisor class group?

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[1]: Note that this differs from some definitions, which consider all ideals instead of only invertible ideals.

[2]: The nonzerodivisor assumption is to rule out easy counterexamples along the lines of $\mathbb{C}[x,y]/(x^2,xy,y^2)$, $I = J = (x,y)$, $IJ=(0)$.

Answer 1

Let $(R,\mathfrak m)$ be a local (not necessarily noetherian) ring, $a\in R$ a non-zero divisor, and $I,J\subset R$ ideals such that $IJ=(a)$. Then $I$ and $J$ are principal.

Write $a=\sum_{i}a_ib_i$ with $a_i\in I$ and $b_i\in J$. Then $I(a^{-1}J)=R$. (Note that $a^{-1}J$ is an $R$-submodule of $Q(R)$, the total ring of fractions of $R$, and this is the frame where the last equation holds.) From $1=\sum_{i}a_i(a^{-1}b_i)$ it follows that $a_i(a^{-1}b_i)\notin\mathfrak m$ for some $i$. Now show that $I=(a_i)$.

Answer 2

In a local domain, every invertible ideal is principal. The proof of this fact can be found in Kaplansky's book "Commutative rings", page 37, Theorem 59.

Answer 3

I needed this statement recently in a form which was slightly more general than those presented in the other answers (in order to avoid dealing with going to stalks in a slightly messy situation). Luckily I found the statement on the Stacks Project at tag 09ME. I'll reproduce the statement and proof here for completeness:

Lemma. Let $A$ be a ring. Let $I,J\subset A$ be nonzero ideals of $A$ such that $IJ=(f)$ for some non-zerodivisor $f\in A$. Then $I$ and $J$ are finitely generated ideals and finite locally free of rank 1 as $A$-modules.

Proof. It suffices to show that $I$ and $J$ are finite locally free $A$-modules of rank 1 by tag 00NX. To do this, write $f=\sum_{i=1}^n x_iy_i$ with $x_i\in I$ and $y_i\in J$. We can also write $x_iy_i=a_if$ for some $a_i\in A$. Since $f$ is a non-zerodivisor, we see that $\sum_{i=1}^n a_i=1$. Thus it suffices to show that $I_{a_i}$ and $J_{a_i}$ are free of rank 1 over $A_{a_i}$. After replacing $A$ by $A_{a_i}$, we conclude that $f=xy$ for some $x\in I$ and $y\in J$. Note that both $x$ and $y$ are non-zerodivisors. We claim that $I=(x)$ and $J=(y)$ which finishes the proof. Namely, if $x'\in I$, then $x'y=af=axy$ for some $a\in A$. Hence $x'=ax$ and we win.