On ideals of product of commutative rings and their applications

Ali Rezaie Aliabad; Foad Obeidavi; Mehdi Badie

REVIEW 3 major objections 6 minor 33 references

On ideals of product of commutative rings and their applications

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For an infinite product of commutative rings, there are at least $2^{2^{|\Lambda|}}$ maximal ideals, and exactly that many when every factor is local.

desk verdict The maximal-ideal counting theorem is likely correct but has a repairable proof gap, and several Section 4 claims are overstated as printed. read the letter →

arxiv 2506.08537 v1 pith:7FUJFEHH submitted 2025-06-10 math.RA math.GN

Mehdi Badie , Ali Rezaie Aliabad , Foad Obeidavi This is my paper

classification math.RAmath.GN MSC 13A1554C40

keywords maximalidealsidealofproductringsfunctionsZariskitopologyultrafilterlocalfilter-idealcorrespondencehull-kernelsets

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 revisits ideals of a product of commutative rings $R = \prod_{\lambda\in\Lambda} R_\lambda$ using filters on the index set. It proves that an infinite product has at least $2^{2^{|\Lambda|}}$ maximal ideals, one for each ultrafilter on $\Lambda$, and exactly that many when every factor ring is local. It also studies the Zariski topology of maximal ideals: each factor's maximal spectrum embeds as a closed subset, the spectrum is disconnected exactly when the ring is a direct sum of two proper ideals, and for every ring $R$ the real-valued continuous functions on $Max(R)$ are isomorphic to those on $Max(C(Y))$ for a compact $T_4$ space $Y$. The final part shows that the hull-kernel sets $h_M(x)$ obey zero-set-like rules, giving algebraic characterizations of regular rings and related classes.

What carries the argument

The filter-ideal construction $i(F,I)=\{x\in R : \exists F\in F,\ x e_{F^c}\in I\}$ and its inverse filter $Z(E(I))=\{Z\subseteq\Lambda : e_Z\in I\}$ form the core mechanism. Here $e_Z$ is the idempotent with coordinate set $Z$. The identity $Z(E(i(F,\prod_\lambda I_\lambda)))=F$ for proper component ideals turns filters on the index set into ideals of the product, and Proposition 3.9(a) turns ultrafilters into maximal ideals. The local-ring equality uses $Z(E(M))$ to reconstruct $M$: an element lies in $M$ exactly when the set of coordinates at which it avoids the local maximal ideal belongs to the ultrafilter $Z(E(M))$.

What would settle it

Take $R=\prod_{n\in\mathbb{N}} F_2$, the product of countably many copies of the two-element field, and list maximal ideals as $M_U=\{x\in R:\{n:x_n=0\}\in U\}$ for ultrafilters $U$ on $\mathbb{N}$. If any maximal ideal of $R$ fails to have this form, or if $|Max(R)|\ne 2^{2^{\aleph_0}}$, the equality claim for local rings is false. The local-ring proof can also be checked directly by verifying, for each maximal ideal $M$, that $x\in M$ exactly when $\{n:x_n\in M_n\}\in Z(E(M))$.

Watch

Extended reading notes

Core claim

Theorems 3.10 and 3.11 give the paper's central count. For any infinite family $\{R_\lambda\}_{\lambda\in\Lambda}$ of commutative rings, $|Max(\prod_\lambda R_\lambda)| \ge 2^{2^{|\Lambda|}}$; if every $R_\lambda$ is local, equality holds. The lower bound is produced by a one-to-one map from ultrafilters on $\Lambda$ into maximal ideals, sending an ultrafilter $U$ to $i(U,\prod_\lambda M_\lambda)$ where $M_\lambda$ is the unique maximal ideal of the local factor; the upper bound is obtained by showing that every maximal ideal of the product arises this way. Alongside this, the paper shows each $Max(R_\lambda)$ is homeomorphic to a closed subset of the product spectrum, and that finite products have spectra homeomorphic to the disjoint union of the factor spectra.

Load-bearing premise

The equality for local rings depends on assuming that every maximal ideal is exactly the set of elements whose 'bad coordinates' belong to the filter of idempotents contained in that ideal; this identification is asserted in the proof rather than derived.

Editorial extensions

If this is right

Every nonempty family of factors contributes a copy of its maximal spectrum inside the product spectrum, so properties invariant under closed embeddings pass from the product to the factors.
For finite products, $Max(\prod_{i=1}^n R_i)$ is homeomorphic to $\bigsqcup_{i=1}^n Max(R_i)$, giving a direct product formula for rings of continuous functions on these spectra.
The disconnectedness criterion 'spectrum disconnected iff the ring is a direct sum of two proper ideals' yields a simple obstruction: if every infinite family of maximal ideals has zero intersection, the ring cannot be such a direct sum.
For each ring $R$, the ring $C(Max(R))$ of real-valued continuous functions is realized as $C(Max(C(Y)))$ for some compact $T_4$ space $Y$, so the function-ring side of the theory reduces to classical rings of continuous functions.
The hull-kernel sets $h_M(x)$ obey the zero-set dictionary: they detect zero elements, units, annihilators, and regularly generated ideals, yielding characterizations of von Neumann regular rings and rings without proper regular ideals.

Reading between the lines

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

The same filter-ideal correspondence is likely to classify prime ideals, not just maximal ideals, in products of local rings: the paper already shows pseudoprime ideals induce ultrafilters, and the two-sided construction is natural on prime spectra.
A direct test case is the product of countably many copies of the two-element field: the theorem predicts $2^{2^{\aleph_0}}$ maximal ideals, one per ultrafilter on the natural numbers.
The $h_M(x)$-as-zero-sets dictionary suggests that zero-set topological notions from $C(X)$ theory, such as P-spaces and almost P-spaces, have algebraic counterparts in arbitrary commutative rings; Corollary 4.20 already gives one such counterpart in terms of regular ideals.

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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

Maxime — the headline result here is the cardinality theorem for maximal ideals of infinite products: |Max(∏Rλ)| ≥ 2^{2^{|Λ|}}, with equality when all Rλ are local. I believe that theorem is true, and it's the part of the paper worth keeping. The construction i(F,I) and the filter Z(E(I)) are clean, and the proof that i(U,∏Mλ) is maximal (Prop 3.9(a)) is standard but executed properly.

That said, as printed the paper has real problems. The proof of Theorem 3.11 never defines the ultrafilter U it uses, and the line 'Z(E(M)) ⊆ U' is meaningless without it. The stress-test note is right: setting U := Z(E(M)) repairs the argument, so the theorem is almost certainly correct, but the version in the arXiv posting does not contain a valid proof of the equality half. That's a repairable but central gap.

Section 4 needs a more careful pass. Corollary 4.11 (Noetherian UFD implies Max(R) is cofinite) is simply false — k[x,y] refutes it. The abstract's disconnectedness criterion also overreaches: as written it omits the standing semiprimitive assumption and the infinite-Max assumption, and without them the statement is false (e.g., a localization of Z with two maximal ideals, or F×F in the vacuous case). Corollary 4.14 inherits the vacuous-case problem. The hM(x) zero-set part leans on the authors' earlier papers; that self-citation is not a flaw in itself, but it limits the independence of that section.

Who is this for? Someone interested in spectra of infinite products of rings or in the ultrafilter machinery for ideals. The core counting result is a genuine, if modest, extension of known results (Levy–Loustaunau–Shapiro, Tarizadeh–Shirmohammadi), and it deserves referee time. My recommendation: send it to peer review, but only with a request that the authors fix the U gap, correct Corollary 4.11, and restate the topological claims with the needed hypotheses.

Referee Report

3 major / 6 minor

Summary. The paper develops a filter-theoretic framework for ideals of products of commutative rings, introducing the ideal i(F,I) and a filter Z(E(I)) attached to an ideal I. The main algebraic claims are cardinality bounds for the maximal spectrum of an infinite product: Theorem 3.10 asserts |Max(∏ R_λ)| ≥ 2^(2^|Λ|), and Theorem 3.11 asserts equality when every R_λ is local. The paper then studies the Zariski topology on Max(R), proving structure and homeomorphism results, a disconnectedness characterization, and several applications involving the rings C(Max(R)) and the sets h_M(x). The abstract and final sections present these results as unconditional statements, but several require hidden hypotheses or are false as printed.

Significance. If the cardinality result is correctly established, it is a valuable contribution: the lower bound is a clean ultrafilter-based generalization of known results, and the equality statement for local factors would determine the cardinality of the maximal spectrum of arbitrary infinite products of local rings. The lower-bound proof in Theorem 3.10 is essentially sound. The topological applications are of moderate interest, but the paper's disconnectedness characterization is false without a semiprimitive hypothesis, and Corollary 4.11 is false as stated. The z-ideal equivalences in Theorem 4.19 rely heavily on the authors' own preceding work [4], so the novelty there is limited. Overall, the central cardinality idea is promising and likely repairable, but the paper in its present form contains unsupported and incorrect claims that need substantial revision.

major comments (3)

[§3, Theorem 3.11] The equality half of the headline theorem is not proved as printed. The proof uses an ultrafilter U without ever defining it, and the line 'F ∈ Z(E(M)) ⊆ U' is therefore meaningless. The natural repair is to set U := Z(E(M)), which is an ultrafilter by Proposition 3.9(b), and then prove M = i(U,I) as follows: for x ∈ M, let F = {λ : xλ ∈ Mλ}; since Rλ is local, xλ is a unit outside F, so the element r defined by rλ = 0 on F and rλ = xλ^{-1} on F^c satisfies e_F = rx ∈ M, hence F ∈ Z(E(M)) = U; also xe_{F^c} ∈ I, so x ∈ i(U,I). Maximality of M and properness of i(U,I) then force equality. Because this step carries the entire equality assertion, the printed proof is incomplete and the authors must supply the missing definition and argument.
[§4, Corollary 4.11] The claim that a Noetherian U.F.D. has cofinite maximal spectrum is false. For R = k[x,y] over a field k, the infinite family F = {(x, y-a) : a ∈ k} of maximal ideals has intersection k(F) = (x), which is nonzero. By Proposition 4.10, Max(k[x,y]) is therefore not cofinite. The corollary should be restricted to rings for which every nonzero ideal is contained in only finitely many maximal ideals, such as PIDs; the 'Thus if R is a P.I.D.' part is plausible, but the first assertion must be corrected or deleted.
[§4, Theorem 4.13 and Abstract] The disconnectedness characterization 'Max(R) is disconnected if and only if R is direct sum of two proper ideals' is false without the semiprimitive standing assumption of Section 4. Let S = Z \ (2Z ∪ 3Z) and R = S^{-1}Z. Then Max(R) has exactly two maximal ideals, so Max(R) is disconnected, but R is a domain with no nontrivial idempotents and is not a direct sum of two proper ideals. The proof relies on facts (4) and (5), which require Jac(R) = 0. The theorem and the abstract's unconditional 'if and only if' must include the semiprimitive hypothesis explicitly.

minor comments (6)

[Example 3.6(a)] The filter is written as F = {A ⊆ N : 1 ∈ Λ}; it should read '1 ∈ A' (or '1 ∈ N').
[§3, Theorem 3.11] The notation I(U) appears in the proof without definition; it should be i(U,I), the ideal introduced in Section 3.
[§4, Theorem 4.1] The proof contains a corrupted chain 'hM (I) ⊆ hcM (I) ⊆ hcM (I) ⊆ hcM (I) ⊆ hcM (J)' that needs to be rewritten as a clean sequence of inclusions.
[§3, Proposition 3.7] The proof of the reverse implication refers to 'Proposition 3.2', which does not exist in the paper; it should refer to Lemma 3.2(c) or a similar correct statement.
[References] Reference [31] is listed as 'Arxive'; this should be formatted as an arXiv preprint with its identifier and date.
[Abstract] The abstract contains the corrupted token '\ff' in the sentence beginning 'Additionally, we show that Max(R) is disconnected'; it should read 'if'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main cardinality theorems are self-contained, with only minor non-load-bearing self-citations in Section 4.3.

full rationale

The central derivation chain (Theorems 3.10 and 3.11) is not circular. Theorem 3.10 constructs an injection from the set of ultrafilters on the index set into Max(∏Rλ) via U ↦ i(U, ∏Mλ); injectivity is supplied by Lemma 3.5(d) and maximality by Proposition 3.9(a), and the count of ultrafilters is a standard external result. No parameter is fitted to the target cardinality, and the lower bound is not assumed as an input. Theorem 3.11 aims to prove the reverse containment by taking U to be the ultrafilter Z(E(M)); the printed proof has a genuine gap because U is never defined and the line "F ∈ Z(E(M)) ⊆ U" is not meaningful as written. This is a proof-completeness defect, not a circularity: the conclusion is not used as an hypothesis, and the intended argument can be completed by setting U = Z(E(M)). The remaining self-citations occur in secondary material: Theorem 4.19 transfers a block of equivalences from the authors' own [4, Theorem 5.8], and Theorem 4.1 uses [5, Theorem 3.5]. These are real citations to prior work by overlapping authors, but they support the z-ideal and Gelfand-ring parts of the paper rather than the headline maximal-ideal counting theorem, and no step reduces by construction to its own input. The score of 2 reflects only these minor, non-load-bearing self-citations, not a circular derivation of the main results.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on standard ultrafilter cardinality in ZFC and on two domain assumptions introduced in Section 4 that the abstract does not carry: semiprimitivity and infiniteness of Max(R). No fitted parameters or invented entities appear.

assumptions (4)

standard math The number of ultrafilters on an infinite set of cardinal kappa is 2^(2^kappa), via [14, Theorem 9.2].
Used in Theorems 3.10 and 3.11 to convert injectivity of the filter-to-ideal map into a lower bound and then into equality.
domain assumption R is semiprimitive, i.e., Jac(R) = 0.
Section 4 preamble says Max(R) is homeomorphic to Max(R/Jac(R)) and then assumes semiprimitivity; fact (5), h^c(I) subset h(J) iff IJ = 0, needs this. The abstract drops it, making the disconnectedness theorem false.
domain assumption Max(R) is infinite.
Section 4 states 'Henceforth, we assume that Max(R) is infinite' before Theorems 4.13 through 4.16. The abstract's derived corollary is vacuously false for R = F x F without this assumption.
standard math Every element outside the unique maximal ideal of a local ring is a unit.
Used in Theorem 3.11 to build rx with zero set F, a step in the equality proof for products of local rings.

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Pith. "Pith review of On ideals of product of commutative rings and their applications." pith.science (2026). https://pith.science/paper/7FUJFEHH

@misc{pith2026250608537,
  author       = {Pith},
  title        = {Pith review of: On ideals of product of commutative rings and their applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FUJFEHH}},
  note         = {Machine review of arXiv:2506.08537}
}

abstract

In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \{ R_\alpha \}_{\alpha \in A} $ is an infinite family of rings, then $ \left| Max \left( \prod_{\alpha \in A} R_\alpha \right) \right| \geqslant 2^{2^{|A|}} $. Notably, if these rings are local then the equality holds. We establish that $ Max(R_\alpha) $ is homeomorphic to a closed subset of $ Max \left( \prod_{\alpha \in A} R_\alpha \right) $, for each $ \alpha \in A $. Additionally, we show that $ Max(R) $ is disconnected \ff $ R $ is direct summand of its two proper ideals. We deduce that if the intersection of each infinite family of maximal ideals of a ring is zero, then the ring is not direct summand of its two proper ideals. Furthermore, we prove that for each ring $R$, $ C\left(Max(R)\right) $ is isomorphic to $ C\left(Max\left(C(Y)\right)\right) $, for some compact $T_4$ space $Y$. Finally, we explore that $h_M(x)$'s can define roles of zero-sets.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

A. R. Aliabad, M. Badie and S. Nazari, An Extension of z-ideals and z◦-ideals, Hact. J. Math. Sta. 49(1) (2020)

work page 2020

[2] [1]

Abdollahpour, A

E. Abdollahpour, A. R. Aliabad and J. Hashemi, On the homomorphisms of ∩-structure spaces, Categ. Gen. Algebr. Struct. Appl. (2024)

work page 2024

[3] [2]

A. R. Aliabad and M. Badie, Fixed-place ideals in commutative rings, Comment. Math. Univ. Carolin

work page

[4] [3]

A. R. Aliabad and M. Badie, On Bourbaki associated prime divisors of an ideal, Quaest. Math. (2018), 1–22

work page 2018

[5] [5]

A. R. Aliabad, M. Badie and S. Nazari, On commutative Gelfand rings , J. Math. Ext. 16. (2021)

work page 2021

[6] [6]

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra . Reading, 1969

work page 1969

[7] [7]

D. D. Anderson and J. Kintzinger, Ideals in direct products of commutative rings, Bull. Austral. Math. Soc. 77 (2008) 477–483

work page 2008

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T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005

work page 2005