Weak Left Coherence in Rings & Monoids

• Weak left coherence is a finitary condition requiring every finitely generated left ideal to be finitely presented, positioning it between full coherence and no constraint on ideal generation.
• It impacts module theory and homological dimensions by relying on exact sequences in rings and the Howson property in monoids, enabling nuanced classification of algebraic structures.
• The property is preserved under finite direct products and quotients by finitely generated ideals, but may fail with localization and certain semidirect product constructions.

Weak left coherence is a finitary condition relevant in the structure theory of rings and monoids, situated strictly between coherence and the absence of any constraint on the finite generation of ideals. In both commutative ring and monoid contexts, weak left coherence is characterized by a requirement that every finitely generated left ideal is finitely presented. This property has implications for module theory, homological dimensions, and the classification of algebraic structures lacking Noetherian or stronger coherence properties. The concept is pivotal in delineating subtle distinctions among various finiteness conditions, especially in the study of extensions, products, and localization phenomena.

1. Foundational Definitions and Characterizations

For a commutative ring $R$, an $n$-presented module $E$ admits an exact sequence of finitely generated free modules of length $n$. Precisely, $E$ is $n$-presented if there exists

$$F_n \rightarrow F_{n-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0 \rightarrow E \rightarrow 0$$

with each $F_i$ finitely generated and free. The notions $0$-presented (finitely generated) and $1$-presented (finitely presented) are special cases.

A ring is coherent if every finitely generated ideal is finitely presented. In contrast, weak left coherence requires:

• For all ideals $I \subseteq J$ with $J$ finitely presented and $I$ finitely generated, then $I$ is finitely presented (Bakkari et al., 2010).
• Equivalently, for every short exact sequence

$$0 \rightarrow I \rightarrow J \rightarrow J/I \rightarrow 0$$

with $J$ finitely presented and $I$ finitely generated, $I$ is finitely presented.

In monoidal contexts, let $S$ be a monoid. $S$ is weakly left coherent if every finitely generated left ideal is finitely presented as a left $S$-act. This property is equivalent to the combination of:

• The intersection of any two principal left ideals is finitely generated (left ideal Howson property).
• Each left annihilator congruence $\ell(a) = \{(u, v) \in S \times S : au = av\}$ is finitely generated as a left congruence (Gould et al., 12 Jun 2025, Gould et al., 17 Nov 2025).

2. Relationship to Other Finitary Properties

Weak left coherence is strictly intermediate between full coherence and less restrictive finiteness conditions:

• Coherent rings are always weakly coherent.
• If a ring contains a regular element, coherence and weak coherence coincide.
• Classes of coherent rings, weakly coherent rings, and strongly 2-coherent rings are pairwise incomparable; there exist rings that are weakly coherent but not coherent, and strongly 2-coherent but not weakly coherent (Bakkari et al., 2010, Gould et al., 12 Jun 2025).
• In the monoid setting, weak left coherence is not implied by nor implies strong left coherence. For instance, free left Ehresmann monoids of rank at least two are weakly coherent but not left coherent (Gould et al., 12 Jun 2025).

3. Structural Transfer and Closure Properties

The preservation and transfer of weak left coherence under algebraic constructions are nuanced:

• Homomorphic Images: If $R$ is weakly coherent and $I$ is a finitely generated ideal, then $R/I$ remains weakly coherent (Bakkari et al., 2010).
• Localization: Weak left coherence is not stable under localization. In particular, examples exist where a ring is weakly coherent but its localization at certain multiplicative sets is not (Bakkari et al., 2010).
• Finite Direct Products: A finite direct product of rings is weakly coherent if and only if each factor is weakly coherent (Bakkari et al., 2010).
• Trivial Extensions: For a local ring $(A, M)$ and $A$-module $E$ with $ME = 0$, the ring $R = A \ltimes E$ is weakly coherent if either $\dim_{A/M} E = \infty$ or $\dim_{A/M} E < \infty$ and $A$ is weakly coherent (Bakkari et al., 2010).
• Monoid Expansions: Retracts preserve weak left coherence in monoids. In special semidirect products $\mathcal{S}(M) = \mathcal{P}(M) \rtimes M$, weak left coherence in $\mathcal{S}(M)$ is characterized by "left co-ordinated" property in $M$, especially when $M$ is right cancellative (Gould et al., 17 Nov 2025).

4. Explicit Examples and Counterexamples

Several concrete constructions demonstrate the boundaries and subtleties of weak left coherence:

• Non-coherent Weakly Coherent Rings: If $A$ is a coherent local ring and $M$ is a maximal ideal not finitely generated, with $E$ an $A/M$-vector space, the trivial extension $A \ltimes E$ is weakly coherent but not coherent (Bakkari et al., 2010).
• Strong 2-Coherence Without Weak Coherence: Certain integral domains exhibit strong 2-coherence but lack weak coherence due to the presence of regular elements (Bakkari et al., 2010).
• Free Left Ehresmann Monoids: For $\mathrm{FLA}(X)$, every intersection of principal left ideals is finitely generated, and all left annihilator congruences are singly generated, so the monoid is weakly left coherent despite not being coherent for $|X| \geq 2$ (Gould et al., 12 Jun 2025).
• Special Semidirect Products and Groups: If $M$ is a group, $\mathcal{S}(M)$ is a proper inverse monoid and hence weakly left coherent (Gould et al., 17 Nov 2025).
• Semilattices and Failures of Lifting: Semilattices and band expansions provide counterexamples where weak left coherence, or its constituents, do not lift from $M$ to $\mathcal{S}(M)$, despite finite generation in $M$ (Gould et al., 17 Nov 2025).

5. Synthesis: Theoretical Impact and Distinctive Properties

Weak left coherence is a substantially weaker but still robust finitary condition:

• It generalizes the module-theoretic presentation property from coherence to a strictly broader class of rings and monoids.
• Weak left coherence ensures that sufficiently "large" finitely presented objects regulate the finitary properties of embedded finitely generated sub-objects.
• The property is stable under direct products and quotients by finitely generated ideals but demonstrably fails under localization and certain semidirect product expansions.
• For monoids, the equivalence with the conjunction of left ideal Howson and finitely left equated (or, in left abundant monoids, just left ideal Howson) provides a clear combinatorial and congruence-theoretic lens for analysis (Gould et al., 12 Jun 2025, Gould et al., 17 Nov 2025).

6. Open Questions and Further Research Directions

Current investigations address open problems in several domains:

• Characterizing weakly coherent rings via homological dimensions or as special types of $(n,d)$-rings (Bakkari et al., 2010).
• Establishing equivalence of weak left coherence with other finiteness properties (such as finite conductor or valuation) in specialized ring or monoid classes.
• Detecting weak left coherence in polynomial extensions ($R[X]$) or power series constructions ($R[[X]]$), with further study into the extension to Szendrei-type (prefix) expansions in the monoid context (Gould et al., 17 Nov 2025).
• Examining skeleton length constraints for generators of left-annihilator congruences and the implications of uniform boundedness.
• Exploring which subclasses of semidirect expansions—such as those restricted to finite sets or specific combinatorial rules—inherit or reflect weak left coherence or its constituent finitary properties.

These avenues suggest significant structural interplay between weak left coherence and other hierarchical coherence notions; further theoretical development appears promising for both ring and monoid theory.