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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 $x$.

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?


[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 $\mathbb{C}[x,y]/(x^2,xy,y^2)$, $I = J = (x,y)$, $IJ=(0)$.

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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 $I=(a_i)$.

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  • $\begingroup$ This is an extension of Kaplansky's proof to local rings which are not necessarily integral domains. $\endgroup$ – user26857 Nov 9 '15 at 10:25
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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.

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