--- title: Weak Left Coherence in Rings & Monoids url: https://www.emergentmind.com/topics/weak-left-coherence type: topic --- # Weak Left Coherence in Rings & Monoids 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 [1003.3040]. - 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 [2506.11321, 2511.13435]. ## 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 [1003.3040, 2506.11321]. - 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 [2506.11321]. ## 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 [1003.3040]. - **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 [1003.3040]. - **Finite Direct Products**: A finite direct product of rings is weakly coherent if and only if each factor is weakly coherent [1003.3040]. - **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 [1003.3040]. - **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 [2511.13435]. ## 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 [1003.3040]. - **Strong 2-Coherence Without Weak Coherence**: Certain integral domains exhibit strong 2-coherence but lack weak coherence due to the presence of regular elements [1003.3040]. - **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 \(|X| \geq 2\) [2506.11321]. - **Special Semidirect Products and Groups**: If \(M\) is a group, \(\mathcal{S}(M)\) is a proper inverse monoid and hence weakly left coherent [2511.13435]. - **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 \(M\) [2511.13435]. ## 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 [2506.11321, 2511.13435]. ## 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 [1003.3040]. - 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 [2511.13435]. - 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. Source: https://www.emergentmind.com/topics/weak-left-coherence