Some new module-theoretic characterizations of $S$-coherent rings

Xiaolei Zhang

REVIEW 4 major objections 4 minor 24 references

Some new module-theoretic characterizations of $S$-coherent rings

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a commutative ring with multiplicative subset S is S-coherent if and only if every s-pure quotient of an s-absolutely pure module—and every direct limit of absolutely pure modules—is again s-absolutely pure.

desk verdict S-pure framework is a genuinely new angle on S-coherent rings, but the converse of Theorem 4.4 uses Lemma 2.5 in the wrong direction and needs a real fix before acceptance. read the letter →

arxiv 2411.14142 v3 pith:NETTM4G4 submitted 2024-11-21 math.AC

Xiaolei Zhang This is my paper · ORCID

classification math.AC MSC 16U2013E0516E50

keywords s-pureexactsequences-absolutelypuremoduleS-coherentringS-flatuniformlyS-torsionabsolutelydirectlimitChasetheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces $s$-pure exact sequences and $s$-absolutely pure modules, uniformly $S$-torsion analogues of the classical pure exact sequences and absolutely pure (FP-injective) modules. The main theorem (Theorem 4.3) asserts that a commutative ring $R$ with multiplicative subset $S$ is $S$-coherent if and only if every $s$-pure quotient of an $s$-absolutely pure module is $s$-absolutely pure, if and only if every pure quotient of an absolutely pure module is $s$-absolutely pure, and if and only if every direct limit of absolutely pure modules is $s$-absolutely pure. This extends the classical characterization of coherent rings in terms of pure quotients of absolutely pure modules to the $S$-localized setting. The paper also proves that $R$ is $S$-coherent if and only if $\mathrm{Hom}_R(E,I)$ is $s$-flat for every $s$-absolutely pure module $E$ and injective module $I$.

What carries the argument

The paper's central objects are $s$-pure exact sequences—short exact sequences $0\to A\to B\to C\to 0$ such that tensoring with any finitely presented module $M$ gives a u-$S$-exact sequence $0\to M\otimes A\to M\otimes B\to M\otimes C\to 0$—and $s$-absolutely pure modules, meaning modules $E$ for which, given any finitely presented $N$, some $s\in S$ annihilates $\mathrm{Ext}^1_R(N,E)$. The load-bearing mechanism is Theorem 3.2, which shows that $E$ is $s$-absolutely pure exactly when every short exact sequence beginning with $E$ is $s$-pure, and Proposition 3.5, which shows that $s$-pure submodules of $s$-absolutely pure modules are $s$-absolutely pure. These transfer the classical pure-quotient argument to the $S$-localized setting, with the uniform $S$-version of the five-lemma [19, Theorem 1.2] making the diagram chases in Theorems 4.2 and 4.4 go through.

What would settle it

Compute the connecting maps in the diagrams of Theorems 4.2 and 4.4 for a specific finitely presented module and a product of flat modules, and check whether the uniform $S$-version of the five-lemma yields a uniform $S$-isomorphism; a single failure of [19, Theorem 1.2] in this setting would invalidate the proofs, as would finding a non-$S$-coherent ring whose pure quotients of absolutely pure modules are all $s$-absolutely pure.

Watch

Extended reading notes

Core claim

The central discovery is that $S$-coherence of $R$ is equivalent to the $s$-absolutely pure class being closed under the quotient and limit operations that characterize classical coherence. Specifically, Theorem 4.3 proves that $R$ is $S$-coherent if and only if (i) any $s$-pure quotient of an $s$-absolutely pure module is $s$-absolutely pure, (ii) any pure quotient of an $s$-absolutely pure module is $s$-absolutely pure, (iii) any pure quotient of an absolutely pure module is $s$-absolutely pure, and (iv) any direct limit of absolutely pure modules is $s$-absolutely pure. Theorem 4.4 adds that $S$-coherence is equivalent to the $s$-flatness of $\mathrm{Hom}_R(E,I)$ for $s$-absolutely pure $E$ and injective $I$, and to the $s$-flatness of $\mathrm{Hom}_R(\mathrm{Hom}_R(F,I_1),I_2)$ for $s$-flat $F$ and injective $I_1,I_2$. These results are proved using the uniform $S$-version of the five-lemma cited from [19, Theorem 1.2], together with pushout and direct-limit arguments.

Load-bearing premise

The load-bearing premise is that the uniform $S$-version of the five-lemma stated in [19, Theorem 1.2] is valid in the full generality used here, since the diagram chases in Theorems 4.2 and 4.4 collapse without it.

Editorial extensions

If this is right

If $R$ is $S$-coherent, then every pure quotient of an absolutely pure module is $s$-absolutely pure; in particular, the classical pure-quotient test for coherent rings survives in the $S$-localized setting.
Conversely, verifying just the direct-limit closure for absolutely pure modules (item (5) of Theorem 4.3) is enough to conclude $R$ is $S$-coherent, giving a single-module-class test.
$S$-coherence is equivalent to the $s$-flatness of $\mathrm{Hom}_R(E,I)$ for all $s$-absolutely pure modules $E$ and all injective modules $I$, and equivalently to the iterated Hom condition with $s$-flat modules and injectives.
When $S$ consists of units of $R$, $s$-pure sequences, $s$-absolutely pure modules, and $S$-coherence reduce to their classical counterparts, so the equivalences recover the known characterizations of coherent rings.
The failure of $s$-absolutely pure modules to be closed under products and direct sums (Example 3.7) shows the characterization cannot be simplified to a limit-closure statement for the class itself.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the final equivalence in Theorem 4.4 hinges on the uniform $S$-version of the five-lemma from [19, Theorem 1.2], a fully self-contained proof of that lemma—or a proof that avoids it—would make the homological characterizations independent of the earlier paper's unstated generality conditions.
The characterization suggests measuring how far a non-$S$-coherent ring is from $S$-coherence by the size of the least $s$ that fails for pure quotients of absolutely pure modules; this would define an $S$-indexed invariant analogous to a relative FP-injective dimension.
The distinction between $s$-absolutely pure and uniformly $S$-absolutely pure modules indicates that the uniform analogue of these theorems would be strictly stronger; a natural test is whether the equivalences of Theorem 4.3 remain valid when 's-absolutely pure' is replaced by 'uniformly $S$-absolutely pure' throughout.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuinely new S-relative framework—s-pure exact sequences and s-absolutely pure modules—and the main equivalences in Theorems 4.2 and 4.3 look plausible. But the proof of Theorem 4.4 has a real gap. In (5)⇒(1) the author shows that D = Hom_R(Hom_R(⊕F_i,I1),I2) is s-flat, then infers that ∏F_i is s-flat because it is a pure submodule of D, citing Lemma 2.5. Lemma 2.5 only says an s-pure quotient of an s-flat module is s-flat; it says nothing about submodules. The needed closure under pure submodules is not proved and does not follow from the stated lemma. That step is load-bearing because it is what lets the author conclude that arbitrary products of flat modules are s-flat, which then yields S-coherence. So the converse direction of Theorem 4.4 is unproved as written. The gap may be fixable if s-flat modules are actually closed under pure submodules, but I am not convinced that is true with this definition, and it needs a standalone proof.

What is good: the new definitions (s-pure, s-absolutely pure, s-flat) extend the classical notions naturally, and Example 2.4 shows s-flat is strictly stronger than the earlier S-flat. Theorem 4.3—S-coherence iff pure quotients of absolutely pure modules are s-absolutely pure, etc.—is a clean structural characterization that generalizes Stenström's theorem. The paper is written in the standard diagram-chasing style of the area and is honest about where earlier results are used.

Other concerns: several load-bearing tools come from the author's own prior papers ([19, Theorem 1.2] and [21, Theorem 2.2(4)]) and are not reproved. That is acceptable practice, but it makes the reader trust those results. There are also typos and compressed five-lemma chases, but those are minor.

Who this is for: people working on S-coherent and S-Noetherian rings will want to know these definitions. The paper deserves a serious referee, but a referee should focus on Theorem 4.4 and ask for either a proof that s-flat modules are closed under pure submodules or a revised argument. As written, I would not accept it without revision; if the gap is fixable, it could be a solid contribution.

Referee Report

4 major / 4 minor

Summary. The paper introduces the notions of s-pure exact sequences and s-absolutely pure modules for a commutative ring R with a multiplicative subset S, and uses them to give new characterizations of S-coherent rings. The main results are Theorem 4.2 (S-coherence iff products of flat modules are s-flat), Theorem 4.3 (S-coherence iff certain closure properties of s-absolutely pure modules hold), and Theorem 4.4 (S-coherence iff certain Hom modules are s-flat). The proofs are largely diagram-theoretic and rely on the author's earlier results, especially a 'uniform S-version of the five-lemma' [19, Theorem 1.2].

Significance. If the main theorems are correct, the paper generalizes classical characterizations of coherent rings due to Chase and Stenström to the S-relative setting, and the new notions of s-pure exact sequences and s-absolutely pure modules are natural additions to the toolkit of S-homological algebra. The paper contains several carefully written proofs, including the equivalence of the various characterizations of s-pure exact sequences in Theorem 2.2 and the characterization of s-absolutely pure modules in Theorem 3.2. However, the proof of the converse direction of Theorem 4.4 contains a significant gap, and the proof of Theorem 4.3 is compressed at a load-bearing point, so the manuscript needs substantial revision before its claims can be accepted.

major comments (4)

[§4, Theorem 4.4, (5)⇒(1)] The proof applies Lemma 2.5 in the wrong direction. After establishing that D = ∏ Hom_R(Hom_R(F_i,I_1),I_2) is s-flat, the text says: 'Since ∏F_i is a pure submodule of ∏ Hom_R(Hom_R(F_i,I_1),I_2), we have ∏F_i is s-flat by Lemma 2.5.' But Lemma 2.5 states that an s-pure quotient of an s-flat module is s-flat; it does not state that a pure submodule of an s-flat module is s-flat. For the exact sequence 0→∏F_i→D→D/∏F_i→0, Lemma 2.5 would apply to the quotient, not to the submodule. The paper gives no other argument for closure of s-flat modules under pure submodules; such a closure is not implied by the stated lemma, and the Tor long exact sequence would require uniform S-torsion of Tor_2(M,D/∏F_i), which is not supplied. Since this step is what yields s-flatness of products of flat modules, the converse direction of Theorem 4.4 is not proved as written.
[§4, Theorem 4.2 and Theorem 4.4, (1)⇒(2)] The proofs of the forward directions of Theorems 4.2 and 4.4 are sketched rather than written out. In both cases, the text says that the conclusion follows from the 'uniform S-version of five-lemma' [19, Theorem 1.2] after chasing diagrams, but the lemma is neither stated nor verified against the hypotheses of the present situation. The paper also invokes [21, Theorem 2.2(4)] and [24, Theorem 7] without stating what these results give. Since these are self-citations of the author's own prior work, the reader cannot check the key diagram chase without consulting external sources. The paper should state the needed lemma and either reproduce the chase or give a precise reference with a verification that its hypotheses hold at the point of use.
[§4, Theorem 4.3, (5)⇒(1)] The proof that every finitely generated ideal is S-finitely presented is too compressed at the point where a map to the direct limit is factored through a stage. After obtaining β: R → lim E(M_i) and factoring it as R → E(M_j), the text says: 'Since the composition I → R → E(M_j) → E(M_j)/M_j becomes to be 0 in the direct limit, we can assume I → R → E(M_j) can factor through some I → M_j.' This is not justified: a map being zero in the direct limit does not imply it is zero at the j-th stage, nor does it imply that the image of I under the factored map lands in M_j. The argument needs an explicit direct-limit chase, possibly after passing to a larger index and modifying β_j by an element of lim M_i. As written, the claimed factorization through M_j is unsupported, and this is the step that produces the u-S-epimorphism lim Hom_R(I,M_i) → Hom_R(I, lim M_i).
[§4, Theorem 4.3, (4)⇒(5)] In the proof of (4)⇒(5), the line 'Note that ⊕M_i is absolutely pure, so is lim M_i by (2)' cites statement (2) of Theorem 4.3, which concerns s-pure quotients of s-absolutely pure modules. The direct limit is a pure quotient of the direct sum, so the relevant statement is (4) (or (3)), not (2). The fact that a direct sum of absolutely pure modules is absolutely pure is standard, but it should be stated or cited explicitly.

minor comments (4)

[§2, Theorem 2.2, proof of (1)⇒(2)] In the sentence beginning 'Now assume that there exists b_j ∈ B', the phrase 'for any j = 1,...,m' should read 'for any i = 1,...,n', since it refers to the indices of the equations f(a_i) = Σ r_{ij} b_j.
[Introduction, Theorem 4.4 statement] In the Introduction's summary of Theorem 4.4, the phrase 'if and only if if I is an injective cogenerator' contains a duplicated 'if'; this should be corrected.
[§4, Theorem 4.4, proof of (5)⇒(1)] The notation in the chain '∏ Hom_R(Hom_R(F_i,I_1),I_2) ∼= Hom_R(⊕ Hom_R(F_i,I_1),I_2)' is confusing: the product on the left should be indexed explicitly (∏_i) or replaced by Hom_R(⊕_i Hom_R(F_i,I_1),I_2), and the isomorphism should be explained as the standard adjunction between Hom and direct sums.
[General] The paper repeatedly uses standard facts without citation or proof, including the purity of the double-dual evaluation map F → Hom_R(Hom_R(F,I_1),I_2) for an injective cogenerator I_2, and the isomorphism Hom_R(⊕M_i,I) ≅ ∏ Hom_R(M_i,I). These facts are standard, but given that they are load-bearing in Theorem 4.4, the author should state them explicitly to aid verification.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's equivalences are derived from its new definitions and independent auxiliary lemmas, with one non-circular proof gap noted in Theorem 4.4.

full rationale

The central characterizations of S-coherent rings in Theorems 4.2, 4.3, and 4.4 are proved from the paper's own definitions of s-pure sequences and s-absolutely pure modules, together with auxiliary results drawn from prior work. The self-citations are numerous but they supply general tools, not the target conclusions: [24, Theorem 7] gives the standard kernel-S-finite consequence of S-coherence, [19, Theorem 1.2] is a uniform-S five-lemma, [21, Theorem 2.2(4)] converts u-S-finite presentation data into S-finite kernels, and [14, Lemma 4.2] is a standard Ext-Tor Hom adjunction. None of these cited results assumes that products of flat modules are s-flat or that direct limits of absolutely pure modules are s-absolutely pure. The equivalences in Theorem 4.3 are obtained by direct implication arguments from Definitions 2.1 and 3.1, Theorem 2.2, Proposition 3.5, and Theorem 3.2; the step (5) implies (1) uses the direct-limit hypothesis to prove every finitely generated ideal is u-S-finitely presented and then applies an independent prior lemma to conclude S-finiteness. Theorem 4.4 similarly combines Proposition 3.4 with Theorem 4.2. One flagged non-circular gap is that in the proof of Theorem 4.4, (5) implies (1), the statement 'Since ∏F_i is a pure submodule of ∏ Hom_R(Hom_R(F_i,I_1),I_2), we have ∏F_i is s-flat by Lemma 2.5' invokes Lemma 2.5 in the converse direction: Lemma 2.5 only says an s-pure quotient of an s-flat module is s-flat, not that an s-pure submodule is s-flat. This is a correctness concern for that converse direction, but it is not circularity, because the conclusion is not assumed or definitionally identical to the cited lemma. Overall, no self-definitional reduction, no fitted-input-called-prediction step, and no load-bearing circular self-citation chain was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted. The paper introduces new mathematical definitions, which are the content of the paper rather than unsupported postulates. The main burden is a set of cited technical results from the author's earlier work; these are the axioms the reader must take on faith.

assumptions (5)

domain assumption R is a commutative ring with identity and S is a multiplicative subset of R
Stated in the introduction; all definitions and theorems depend on this setting.
standard math Every R-module has an injective envelope
Used in Theorem 3.2 and Theorem 4.3 for the modules E(M_i). This is a standard ZFC consequence (Baer, injective hulls exist).
standard math Uniform S-version of the five-lemma ([19, Theorem 1.2]) is valid and applicable to the diagrams in Theorems 4.2 and 4.4
The paper invokes it without proof to conclude that certain maps are uniformly S-isomorphisms; this is the main external tool.
standard math For an S-coherent ring, the kernel of a finitely generated presentation of a finitely presented module is S-finitely presented ([24, Theorem 7])
Used in Theorem 4.2 (1) to (2) to obtain the exact sequence 0 to K to Q to N to 0.
standard math A u-S-finitely presented module has an S-finite kernel ([21, Theorem 2.2(4)])
Used at the end of Theorem 4.3 (5) to (1) to conclude that the ideal I is S-finitely presented.

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Pith. "Pith review of Some new module-theoretic characterizations of $S$-coherent rings." pith.science (2026). https://pith.science/paper/NETTM4G4

@misc{pith2026241114142,
  author       = {Pith},
  title        = {Pith review of: Some new module-theoretic characterizations of $S$-coherent rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NETTM4G4}},
  note         = {Machine review of arXiv:2411.14142}
}

abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we first introduce and study the notions of $s$-pure exact sequences and $s$-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then, we give some new characterizations of $S$-coherent rings in terms of $s$-absolutely pure modules.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

D. D. Anderson, T. Dumitrescu, S-Noetherian rings , Comm. Algebra 30 (2002), 4407-4416

work page 2002

[2] [2]

Angeleri H¨ ugel, D

L. Angeleri H¨ ugel, D. Herbera, Mittag-Leﬄer conditions on modules , Indiana Math. J. 57, (2008), 2459-2517

work page 2008

[3] [3]

Bazzoni, L

S. Bazzoni, L. Positselski, S-almost perfect commutative rings, J. Algebra 532 (2019), 323-356

work page 2019

[4] [4]

S-FP-injective modules

D. Bennis, A. Bouziri, S-FP-injective modules, http://arxiv.org/abs/2410.00167

work page Pith review arXiv

[5] [5]

Bennis, M

D. Bennis, M. El Hajoui, On S-coherence, J. Korean Math. Soc. 55 (2018), no. 6, 1499-1512. 16

work page 2018

[6] [6]

J. L. Chen, N. Q. Ding, The weak global dimension of commutative coherent rings , Comm. Algebra 21(1993), no.10, 3521-3528

work page 1993

[7] [7]

Fuchs, Abelian Groups (Springer, Switzerland, 2015)

L. Fuchs, Abelian Groups (Springer, Switzerland, 2015)

work page 2015

[8] [8]

Fuchs, L

L. Fuchs, L. Salce, Modules over Non-Noetherian Domains , Providence, AMS, 2001

work page 2001