Reconstruction of Krull monoids from conductor submonoids
Henry Zweiman
September 9, 2026
Abstract
Let be a divisor theory of a commutative Krull monoid, and let . We show that determines the reduced monoid of , together with its divisor theory. The reconstruction takes the complete integral closure and then generates a submonoid from its divisibility-minimal nonunits. Consequently two such conductor monoids are isomorphic exactly when the original reduced Krull monoids are isomorphic, answering an isomorphism question of Geroldinger, Yan, and Zhong. We also describe the automorphism group in terms of the characteristic and give an intrinsic recognition criterion for conductor monoids that arise in this way.
1. The isomorphism question
All monoids in this note are commutative and cancellative, with identity. Write for the group of units, , and for the quotient group. A reduced free abelian monoid consists of the finite products of primes from . A homomorphism is a divisor homomorphism if divisibility of by implies divisibility of by . It is a divisor theory if, in addition, every element of is the gcd of a finite nonempty subset of . It suffices to impose this condition on the primes of . A monoid is Krull precisely when it admits a divisor theory.
Geroldinger, Yan, and Zhong associate to a divisor theory the conductor submonoid
Immediately after Proposition 4.3 of [2], they ask when the monoids associated to two Krull monoids are isomorphic. Their notation is . We use to distinguish it from the ambient free monoid.
Theorem 1.1.
For , let be divisor theories of Krull monoids. There is a natural bijection
It respects identities and composition. In particular, the two conductor monoids are isomorphic if and only if the two reduced Krull monoids are isomorphic.
The complete integral closure of a conductor monoid is already identified in [2, Lemmas 3.1–3.2]. The additional observation needed here is that minimal nonunits in that closure recover the atoms of the original Krull monoid. Thus the proof uses standard divisor theory and is short. The automorphism and recognition statements below are consequences of the same reconstruction, rather than separate isomorphism problems.
2. Intrinsic recovery of the original monoid
For a monoid , its complete integral closure is
An isomorphism extends uniquely to quotient groups and carries complete integral closures onto one another.
Lemma 2.1.
Let be free abelian, and let be a nonempty ideal with . For , one has and .
Proof.
Fix . For every , both and belong to , so . This proves equality of quotient groups. Also for all , proving . If , some prime valuation is negative. For every fixed , the integer is eventually negative. Hence cannot belong to for all , and .
Whenever is free abelian and is reduced, define
This definition is intrinsic to . It uses divisibility in the recovered closure, not divisibility inside .
Lemma 2.2.
Let be a divisor theory of a nontrivial reduced Krull monoid. Then
Proof.
The divisor homomorphism is injective because is reduced. The nonunits of form the nonempty ideal , and Lemma 2.1 identifies the closure with .
Take . Some satisfies . The Krull monoid is atomic, so an atom divides . Since is a nonunit of dividing in , minimality gives .
Conversely, let and suppose that divides in . Choose such that . The divisor homomorphism property gives . Thus , and
forces . This proves the first equality. Atomicity proves the second.
A divisor theory factors through , so we may suppose both original monoids reduced. If , the gcd condition forces ; this case is immediate.
An isomorphism extends uniquely to quotient groups. By Lemma 2.1 it restricts to an isomorphism . This extension preserves divisibility and maps onto . Lemma 2.2 therefore gives a restricted isomorphism of with , and hence one of with .
Conversely, an isomorphism identifies two divisor theories of . The uniqueness theorem for divisor theories [1, Theorem 2.4.7] supplies a unique isomorphism with . Equation (1) shows that its restriction maps onto . Uniqueness makes the two constructions inverse and ensures compatibility with composition.
3. Characteristic and automorphisms
Identify a reduced Krull monoid with its image in a divisor theory . Set
The group and prime multiplicities are the characteristic. The familiar classification of reduced Krull monoids by this data is [1, Theorem 2.5.4]. We record explicitly how it answers the conductor question.
Corollary 3.1.
Two conductor monoids in Theorem 1.1 are isomorphic exactly when there is a group isomorphism with for every .
Proof.
An isomorphism of conductor monoids extends to the free monoids and preserves the recovered submonoids, so it induces the required isomorphism on quotient groups and bijections of prime fibers. Conversely, choose such fiber bijections. The resulting isomorphism of free monoids carries onto . Since a divisor homomorphism identifies
it carries onto , and therefore also the conductor monoids.
Corollary 3.2.
Let . There is a split exact sequence
Thus, after choices of labels on the prime fibers,
The product is unrestricted, including when is infinite.
Proof.
Theorem 1.1 identifies automorphisms with prime permutations preserving , equivalently its quotient-group kernel. The kernel of their action on consists exactly of the independent permutations within the fibers . Every member of lifts by choosing fiber bijections, as in Corollary 3.1.
For a splitting, choose, for each cardinal occurring among the multiplicities, a fixed set and bijections . Lift by . These lifts compose and preserve the quotient-group kernel. Every permutation of an infinite prime set still acts on its free abelian monoid, since each monomial has finite support.
For a Dedekind domain, the associated conductor monoid can be described as the unit ideal together with the nonzero ideals contained in a proper principal ideal [2, Example 4.4]. The reconstruction recovers the monoid of nonzero principal ideals and its divisor theory. It recovers the multiplicative monoid modulo units, not the addition law or the unit group of the domain.
4. Recognition and examples
The construction does not exhaust all conductor submonoids. The following criterion gives an intrinsic test.
Proposition 4.1.
Let , where is free abelian and is a nonempty proper ideal. Put . Then is isomorphic to the conductor monoid of a divisor theory of a Krull monoid if and only if
;
every prime of is the gcd of a finite nonempty subset of .
When these conditions hold, the inclusion is the recovered divisor theory and .
Proof.
For every , choose a divisor of in of least free-monoid length. It lies in . Hence . The two stated conditions say respectively that the inclusion is a divisor homomorphism and that it satisfies the gcd condition. Thus they make it a divisor theory, with Krull. Since , its associated conductor monoid is exactly .
Conversely, any isomorphism with a conductor monoid of a divisor theory extends to the free closures. Lemma 2.2 identifies with the image of that Krull monoid, proving both conditions.
If has finite rank, is a finite antichain of exponent vectors. To check condition (2), it suffices, for each prime , to take the gcd of
and require it to be . Indeed, if divides a product of elements of , it divides at least one factor. The gcd of all -divisible elements of therefore equals the gcd of . Condition (1) is the equality of the affine monoid generated by these vectors with the intersection of their integer span and the nonnegative orthant. This formulation separates saturation from the indispensable gcd condition.
Example 4.2.
In additive notation, let , , and
The minimal elements are exactly the vectors of coordinate sum . Their generated monoid is
To see the equality, repeatedly remove a nonnegative subvector of sum . Its quotient group consists of the integer vectors whose coordinate sum is divisible by : differences of degree- vectors include , and is present. Thus condition (1) holds. For , the coordinatewise minimum of and , with , is . For , itself is present. This verifies condition (2).
The recovered class group is , with all primes in the class of . Its only automorphism preserving the occupied class fixes and is the identity. Consequently . The class group recovered from the conductor monoid distinguishes the parameter , even though this automorphism group does not.
Example 4.3.
For , the rank-one conductor monoid has and . It satisfies , but no gcd of nonzero elements of is . Proposition 4.1 shows that is not isomorphic to the conductor monoid of any divisor theory of a Krull monoid. An arbitrary divisor homomorphism cannot replace a divisor theory in Theorem 1.1.
References
[1] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC, Boca Raton, 2006.
[2] A. Geroldinger, W. Yan, and Q. Zhong, On conductor submonoids of factorial monoids, Semigroup Forum112 (2026), 792–819. doi:10.1007/s00233-025-10583-6.