Ideal Class Monoid: Structure & Computation

• Ideal class monoid is the quotient of fractional ideals by principal ideals, capturing both invertible and non-invertible cases.
• It stratifies by overorders using multiplicator rings to reveal local decompositions and the breakdown of Dedekind behavior.
• Computational algorithms classify weak equivalence classes locally, linking the structure to density computations and Drinfeld module isogeny counts.

Searching arXiv for recent and foundational papers on ideal class monoids and closely related formulations. The ideal class monoid is the quotient of fractional ideals by principal ideals in contexts where ideal multiplication still makes sense but ideal inversion may fail. For an order $R$ with total quotient ring $K$, it may be written as the set of isomorphism classes of fractional $R$-ideals under scaling by $K^\times$; its unit group is the class group of invertible ideals, while its non-units record the presence of non-invertible fractional ideals and, therefore, the failure of Dedekind behavior (Eggink, 19 Jan 2026, Marseglia, 2018, Cho et al., 26 Sep 2025). In current literature the notion appears in local function-field arithmetic, in the arithmetic of non-maximal orders in number fields, in the class semigroup of infinite integral extensions of Dedekind domains, and in combinatorial settings such as numerical semigroups (Eggink, 19 Jan 2026, Ohshita, 2018, Bonzio et al., 2024).

1. General definition and algebraic structure

For an order $R$, the ideal class monoid is the set of fractional $R$-ideals modulo principal ideals, with multiplication induced by ideal multiplication (Eggink, 19 Jan 2026). In the language of orders in finite products of number fields, one writes

$\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$

where $\mathcal{I}(R)$ is the monoid of fractional $R$-ideals and $\mathcal{P}(R)$ is the set of principal fractional ideals $K$0 with $K$1 (Marseglia, 2018). In the number-field literature the notation $K$2 is also used for the monoid of equivalence classes of all fractional $K$3-ideals modulo multiplication by $K$4, while $K$5 denotes the subgroup of classes of invertible fractional ideals (Cho et al., 26 Sep 2025).

The distinction between monoid and group is fundamental. In a Dedekind domain every nonzero ideal is invertible, so the quotient by principal ideals is a group. For a non-maximal order, non-invertible fractional ideals generally occur, and the quotient is only a commutative monoid (Eggink, 19 Jan 2026). The subgroup of units of this monoid is precisely the Picard group, or ideal class group of invertible ideals (Marseglia, 2018, Cho et al., 26 Sep 2025).

A central organizing device is the multiplicator ring

$K$6

For a fractional ideal $K$7, this is an overorder of $K$8, and it is the largest overorder over which $K$9 is again a fractional ideal (Eggink, 19 Jan 2026, Marseglia, 2018). This yields a decomposition of the ideal class monoid into strata indexed by overorders. In the notation of the local function-field setting,

$R$0

and

$R$1

with the union over all overorders $R$2 of $R$3 (Eggink, 19 Jan 2026). In the global number-field setting the same principle appears as a stratification of $R$4 by overorders and of the orbit set $R$5 by the same data (Cho et al., 26 Sep 2025, Cho et al., 8 Jul 2026).

2. Local orders in the function-field setting

A particularly explicit formulation occurs in the function-field setting of $R$6, with $R$7 a nonzero prime ideal, $R$8 irreducible of degree $R$9, and

$K^\times$0

The local problem is posed at the $K^\times$1-adic completion $K^\times$2 (Eggink, 19 Jan 2026). If $K^\times$3 are the primes of $K^\times$4 above $K^\times$5, then

$K^\times$6

and each factor $K^\times$7 is a complete local ring (Eggink, 19 Jan 2026).

In this setting, a fractional $K^\times$8-ideal is a nonzero finitely generated $K^\times$9-submodule of

$R$0

that becomes an ordinary ideal after multiplication by some nonzero element (Eggink, 19 Jan 2026). The local ideal class monoid is then

$R$1

with multiplication

$R$2

Because $R$3 is generally a non-maximal order, non-invertible fractional ideals may occur, so this quotient is a commutative monoid rather than a group (Eggink, 19 Jan 2026).

The local order $R$4 is studied because several arithmetic counting problems factor over primes. The paper "Calculating The Local Ideal Class Monoid and Gekeler Ratios" (Eggink, 19 Jan 2026) states that $R$5 captures the local endomorphism structure relevant to counting matrices with prescribed characteristic polynomial and to computing weighted sizes of isogeny classes of Drinfeld modules. This makes the local ideal class monoid a local parameter space for rank-one module structures over $R$6 (Eggink, 19 Jan 2026).

3. Overorders, weak equivalence, and the local decomposition

The local structure of $R$7 is controlled by overorders and by weak equivalence classes. An overorder $R$8 of $R$9 is an intermediate ring

$R$0

that is finite over $R$1 (Eggink, 19 Jan 2026). Every multiplicator ring $R$2 of a fractional $R$3-ideal is such an overorder.

Weak equivalence is the local relation that identifies ideals differing by invertible ideal multiplication inside a common multiplicator ring. For fractional $R$4-ideals $R$5 and $R$6, the following are equivalent (Eggink, 19 Jan 2026): $R$7

$R$8

and

$R$9

for some invertible fractional $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$0-ideal $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$1. When these conditions hold one writes $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$2, and the set of weak equivalence classes with fixed multiplicator ring $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$3 is denoted

$\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$4

A decisive simplification occurs locally: the Picard group of $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$5 is trivial,

$\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$6

because

$\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$7

and each factor $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$8 is local, so every invertible ideal is principal (Eggink, 19 Jan 2026). Consequently, for every local overorder $\mathrm{ICM}(R)=\mathcal{I}(R)/\mathcal{P}(R),$9,

$\mathcal{I}(R)$0

and therefore

$\mathcal{I}(R)$1

This identifies the local ideal class monoid with the disjoint union of local weak equivalence classes (Eggink, 19 Jan 2026).

A useful extreme case is that of a regular prime. If every local factor $\mathcal{I}(R)$2 is a discrete valuation ring, then every ideal is principal and $\mathcal{I}(R)$3 is trivial (Eggink, 19 Jan 2026). This shows that the nontriviality of the ideal class monoid is concentrated at primes where the local order fails to be DVR-like.

4. Computation of the local ideal class monoid

The paper (Eggink, 19 Jan 2026) proves that $\mathcal{I}(R)$4 is effectively computable when $\mathcal{I}(R)$5 is irreducible and $\mathcal{I}(R)$6. The algorithm is organized around a local-to-global reduction.

First, one computes global $\mathcal{I}(R)$7-overorders $\mathcal{I}(R)$8 of $\mathcal{I}(R)$9, then passes to the corresponding local overorders $R$0. The key structural statements are two bijections: the map from global $R$1-overorders of $R$2 to $R$3-overorders of $R$4, and the bijection between overorders of $R$5 and overorders of $R$6 obtained by completion and intersection with $R$7 (Eggink, 19 Jan 2026). In the language of orders in étale $R$8-algebras, overorders are also precisely the idempotent fractional ideals, so enumeration of overorders is already a central step in the global computation of an order’s ideal class monoid (Marseglia, 2018).

Second, for each global overorder $R$9, one computes the set $\mathcal{P}(R)$0 of global weak equivalence classes of fractional $\mathcal{P}(R)$1-ideals with multiplicator ring $\mathcal{P}(R)$2. The paper states that this uses algorithms developed in earlier work, with a modification in the inseparable case regarding a lemma on trace forms, while the overall algorithm remains valid (Eggink, 19 Jan 2026).

Third, one identifies which global weak equivalence classes merge after localization and completion. The decisive criterion is

$\mathcal{P}(R)$3

together with the equivalence

$\mathcal{P}(R)$4

The map

$\mathcal{P}(R)$5

is well-defined and surjective (Eggink, 19 Jan 2026).

The resulting procedure is summarized in four steps (Eggink, 19 Jan 2026):

1. Compute global $\mathcal{P}(R)$6-overorders $\mathcal{P}(R)$7 of $\mathcal{P}(R)$8, then the corresponding local overorders $\mathcal{P}(R)$9.
2. Compute the global weak equivalence classes $K$00.
3. Determine which global classes merge into each local $K$01.
4. Take the disjoint union of the local weak equivalence classes; this is $K$02.

This algorithm extends the older computational focus on invertible classes to the genuinely monoidal setting that includes non-invertible ideals (Marseglia, 2018).

5. Local densities, Gekeler ratios, and Drinfeld modules

In the function-field application of (Eggink, 19 Jan 2026), the local ideal class monoid feeds into the local Gekeler ratio

$K$03

This is a normalized local density for matrices over $K$04 with characteristic polynomial $K$05 (Eggink, 19 Jan 2026).

The connection with $K$06 is structural. Local conjugacy classes of matrices with characteristic polynomial $K$07 correspond to isomorphism classes of $K$08-modules of rank $K$09 over $K$10, or equivalently to fractional $K$11-ideals up to isomorphism (Eggink, 19 Jan 2026). Different local ideal classes contribute with different automorphism-group sizes, and the decomposition by overorders stratifies the local counting problem according to local endomorphism structure (Eggink, 19 Jan 2026).

The global object is the product

$K$12

which the paper uses as part of an algorithm to compute the weighted size of an isogeny class of Drinfeld modules (Eggink, 19 Jan 2026). In that interpretation:

• local orders $K$13 describe local endomorphism behavior,
• $K$14 parametrizes local rank-one module structures,
• the local Gekeler ratios encode the weighted contribution of these local structures to the global count (Eggink, 19 Jan 2026).

A frequent misconception is that the triviality of the local Picard group should force the local ideal class monoid to be trivial. The local theory shows the opposite: $K$15 eliminates only invertible ambiguity, while non-invertible ideals survive as distinct weak equivalence classes and can make $K$16 highly nontrivial (Eggink, 19 Jan 2026).

6. Broader variants and related theories

The phrase “ideal class monoid” is also used in several adjacent frameworks. For orders in number fields, $K$17 denotes the monoid of classes of all fractional ideals modulo scaling by $K$18. Recent work gives an upper bound for $K$19 in terms of the class number of $K$20 and local orbital integrals for $K$21, and for Gorenstein cubic orders it gives a closed Euler product formula for the cardinality of the genus set

$K$22

with local factors coming from an explicit classification of local cubic overorders (Cho et al., 26 Sep 2025, Cho et al., 8 Jul 2026). In the same cubic setting, that formula yields the exact number of $K$23-equivalence classes of integral $K$24-orbits whose associated cubic ring is a prescribed Gorenstein order $K$25 (Cho et al., 8 Jul 2026).

In the theory of infinite integral extensions of Dedekind domains, the class object becomes a class semigroup

$K$26

where $K$27 is the monoid of fractional ideals and $K$28 the principal fractional ideals. For one-dimensional normal domains arising from infinite integral extensions, fractional ideals admit a description by bounded, compactly supported, upper $K$29-semicontinuous functions on $K$30, and the class semigroup is analyzed through this function-theoretic model (Ohshita, 2018). This shows that the transition from class group to class monoid is not limited to non-maximal orders; it also arises in non-Noetherian normal domains.

In numerical semigroup theory, the ideal class monoid $K$31 is the quotient of ideals by translation. It is isomorphic to the monoid of ideals with smallest element $K$32, and its structure is controlled by Apéry sets and Kunz coordinates (Casabella et al., 2023). Its poset is a lattice precisely when the multiplicity is at most $K$33, and irreducibility of the numerical semigroup is equivalent to the ideal class monoid having at most two quarks (Bonzio et al., 2024). This is a genuinely combinatorial incarnation of the same idea: one keeps a monoidal quotient even though invertibility is absent.

At a more abstract level, ideal systems on cancellative monoids produce a monoid $K$34 of $K$35-invertible $K$36-ideals and a corresponding class group

$K$37

so radical factorization and $K$38-almost Dedekind conditions can be expressed directly in the arithmetic of ideal monoids (Olberding et al., 2019). This suggests that the ideal class monoid is best viewed not as a single specialized construction, but as a general mechanism for retaining multiplicative ideal theory after group structure has broken down.