REVIEW 6 minor 19 references
On the prime spectrum of the amalgamations
T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For an amalgamated ring $R\bowtie^fJ$, the pm property — every prime ideal lying in exactly one maximal ideal — is characterized by a two-term count of maximal ideals equal to one for each prime $q$ of $S$ avoiding $J$, together with $R$…
desk verdict A clean, correct characterization of pm-rings among amalgamations that fixes a published error; the transfer results in Section 4 have minor gaps but are sound. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the amalgamated algebra $R\bowtie^fJ$, with its two families of prime ideals: type 1 primes $p'^f=\{(p,f(p)+j): p\in p,\ j\in J\}$ indexed by $p\in\operatorname{Spec}(R)$, and type 2 primes $q^f=\{(r,f(r)+j): f(r)+j\in q\}$ indexed by $q\in\operatorname{Spec}(S)\setminus V(J)$. The argument is carried by the inclusion rules (Lemma 2.4): type 1 inclusions mirror inclusions in $R$; type 2 inclusions mirror inclusions in $S$; a type 2 prime $q^f$ lies inside a type 1 prime $p'^f$ exactly when $f^{-1}(q+J)\subseteq p$; and a type 1 prime is never contained in a type 2 prime. Theorem 3.4 counts the maximal ideals above $q^f$ by applying these rules to maximal ideals, which turns the pm property into the displayed equality.
What would settle it
A concrete way to test the theorem is to compute the two displayed cardinalities for every $q\in\operatorname{Spec}(S)\setminus V(J)$ in the localized polynomial-ring example constructed in the paper: with $R=k$ a field, $S=T^{-1}k[x,y]$ localized away from $\langle x\rangle$ and $\langle y\rangle$, $J=T^{-1}\langle x\rangle$, the two terms are 1 and 1 for $q=0$, so the equality fails and the amalgamation is not a pm-ring. The theorem would be refuted by any instance where the equality holds for every such $q$ yet some prime ideal of $R\bowtie^fJ$ is contained in two distinct maximal ideals; searching small finite rings for such a pair is a direct falsification check.
Extended reading notes
Core claim
The central claim is Theorem 3.4. For a ring homomorphism $f:R\to S$ and a nonzero proper ideal $J$ of $S$, write $R\bowtie^fJ=\{(r,f(r)+j): r\in R,\ j\in J\}$. The paper proves that $R\bowtie^fJ$ is a pm-ring if and only if $R$ is a pm-ring and, for every $q\in\operatorname{Spec}(S)\setminus V(J)$, $$\left|(\operatorname{Max}(S)\cap V(q))\setminus V(J)\right|+\left|\operatorname{Max}(R)\cap V($f^{{-1}}$(q+J))\right|=1.$$ The proof runs through the known description of the prime spectrum: every prime ideal of the amalgamation is either of type 1, $p'^f$, coming from a prime $p$ of $R$, or of type 2, $q^f$, coming from a prime $q$ of $S$ avoiding $J$, and maximal ideals are the same with primes replaced by maximal ideals. For a type-2 prime $q^f$, the two terms in the count list the containing maximal ideals of each type, and the pm condition forces exactly one of them. The paper also exhibits a concrete example where both terms equal 1 and the amalgamation is not a pm-ring, showing why the earlier "if" direction of a prior characterization is invalid.
Load-bearing premise
The whole criterion depends on the previously established description of $\operatorname{Spec}(R\bowtie^fJ)$ as exactly the type-1 and type-2 primes, with the containment rules listed above; if that classification omitted any prime or misdescribed an inclusion, the counting equality would not capture all maximal ideals above $q^f$.
Editorial extensions
If this is right
- The amalgamated duplication $R\bowtie I$ along an ideal $I$ is a pm-ring if and only if $R$ is a pm-ring (Corollary 3.5).
- If $J$ lies in the Jacobson radical of $S$, in particular for Nagata's idealization or the trivial extension $R\ltimes M$, the amalgamation is a pm-ring exactly when $R$ is (Corollary 3.6).
- An earlier characterization of pm-amalgamations is corrected: its "if" direction is false, as shown by a counterexample, and one of its corollaries about h-local rings fails (Remark 3.3).
- If an amalgamation is compactly packed, then both $R$ and $\operatorname{Spec}(S)\setminus V(J)$ are compactly packed; under $J\subseteq\operatorname{Nil}(S)$ the property transfers back (Theorem 4.2, Corollary 4.4).
- For surjective $f$, the amalgamation is properly zipped if and only if $R$ is; the surjectivity assumption is not redundant (Theorem 4.7, Example 4.10).
Reading between the lines
- The paper does not pursue it, but the same maximal-ideal count could be used to test other local-type properties of amalgamations, such as being h-local, since that property also fails exactly when some prime has more than one containing maximal ideal.
- Because the equality involves only finitely many primes in a semilocal setup, the criterion turns the pm question into a finite computation for rings whose spectra are finite.
- Example 4.10 leaves open what happens for non-surjective $f$ in the properly zipped transfer; the natural next step is to find a hypothesis on $f$ that replaces surjectivity.
- The compactly packed direction is proved only one-way in general; the missing converse may require imposing a compactly packed condition on $\operatorname{Spec}(S)\setminus V(J)$ or a finiteness restriction.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the prime spectrum of the amalgamated ring R ⋈^f J. It proves several containment lemmas relating prime ideals of the amalgamation to those of R and S, then gives a complete characterization of when R ⋈^f J is a pm-ring in Theorem 3.4: the ring is pm if and only if R is pm and, for every q ∈ Spec(S)\V(J), the number of maximal ideals of S containing q but not J plus the number of maximal ideals of R containing f^(−1)(q+J) is exactly 1. The paper also provides Example 3.2, a counterexample to the asserted 'if' direction of [15, Theorem 2.4], and in Section 4 studies transfer of the compactly packed and properly zipped properties to amalgamations, with a full characterization of the properly zipped property under surjective f in Theorem 4.7.
Significance. If Theorem 3.4 is correct, it gives a clean, checkable criterion for the pm property in the broad class of amalgamated algebras, while correcting a previously published theorem. The result is accompanied by useful corollaries for amalgamated duplication and trivial extensions, and by an explicit counterexample that independently validates the new criterion. Section 4 adds transfer results for compactly packed and properly zipped rings, both of which are natural companions to the pm property in the study of prime spectra. The proofs are mostly elementary and transparent, and the main theorem is genuinely useful for constructing rings with prescribed maximal-spectrum behavior.
minor comments (6)
- [Lemma 2.2(5)] The notation in the statement is ambiguous because the subscript α is used both for the prime ideal in p′f_α and as the indexing variable in the union ∪α∈Λ qα^f. Please restate as: for every α, p′f_α ⊈ ∪δ∈Λ qδ^f.
- [Lemma 2.2(3) and Lemma 2.3(4)] The proofs of the inverse inclusions in Lemma 2.2(3) and of Lemma 2.3(4) are omitted with 'similar' or 'left to the reader'. Since these statements are used later, including the short arguments would make the paper more self-contained.
- [Corollary 3.6(1)] The proof that the second term of equality (§) equals 1 needs an explicit existence step: one should note that q+J is a proper ideal of S and hence f^(−1)(q+J) is a proper ideal of R, so it is contained in at least one maximal ideal. As written, the argument only shows that there is at most one such maximal ideal.
- [Theorem 4.7, Case 1] In the subcase where ∩δ∈∆ qδ^f ⊆ p′f, the step from f^(−1)(∩δ qδ + J) ⊆ p to ∩δ f^(−1)(qδ) ⊆ p is valid but should be stated explicitly, and the equality f^(−1)(qδ + J) = f^(−1)(qδ) + f^(−1)(J) under surjectivity should be justified or cited.
- [Example 3.2] The paper refers to the 'if direction' of [15, Theorem 2.4] without stating the theorem. A one-sentence statement of the flawed theorem would help the reader appreciate the correction.
- [Throughout] There are several typographical and formatting issues, including the broken word 'amalgama tions' in the title line, the repeated symbol '⊲ ⊳' with inconsistent spacing, and the unicode artifact '/llbracketX/rrbracket' in the introduction. These should be cleaned up.
Circularity Check
No significant circularity: Theorem 3.4 derives from an external prime-spectrum classification and in-paper inclusion lemmas.
full rationale
The central theorem, Theorem 3.4, is not circular. Its load-bearing inputs are the spectrum classification of R⋈^f J from [10, Proposition 2.6] (D'Anna–Finocchiaro–Fontana, not the present author) and the containment criteria of Lemma 2.4, which are proved in Section 2 directly from the definitions of the two types of prime ideals. The counting condition (§) is not an assumed input; it is derived from the pm-ring definition by enumerating the maximal ideals containing each prime ideal qf, using Lemma 2.4 and Remark 2.1. The converse direction uses the pm property of R together with the same counting condition to verify uniqueness of a maximal ideal over each prime of the amalgam. No fitted parameter is renamed as a prediction, and no result is assumed in the form it is claimed to establish. The self-citations [1]–[4] appear only as context or as references to earlier constructions; in particular, the reference to [1] around Lemma 2.4 points to inclusion facts that are independently proved in the present paper as Lemmas 2.2–2.3, with Lemma 2.4 following immediately from those definitions. The counterexample in Example 3.2, which refutes the earlier theorem [15, Theorem 2.4], provides an independent check on the claimed correction. No circular step can be exhibited by quoting the paper, so the appropriate finding is no significant circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption The prime spectrum of R ⊲⊳^f J is exactly the disjoint union {p′f | p ∈ Spec(R)} ∪ {qf | q ∈ Spec(S)\V(J)}, with the analogous maximal ideal decomposition (Remark 2.1, cited from [10, Proposition 2.6]).
- standard math If A∩B ⊆ P for a prime ideal P, then A ⊆ P or B ⊆ P.
- domain assumption Homomorphic images of properly zipped rings are properly zipped.
- domain assumption A ring with infinitely many maximal ideals whose intersection is zero is not properly zipped (cited from [19, Corollary 4.4]).
Cite this review
Pith. "Pith review of On the prime spectrum of the amalgamations." pith.science (2026). https://pith.science/paper/JL7RU3DI
@misc{pith2026241116888,
author = {Pith},
title = {Pith review of: On the prime spectrum of the amalgamations},
year = {2026},
howpublished = {\url{https://pith.science/paper/JL7RU3DI}},
note = {Machine review of arXiv:2411.16888}
}
abstract
Let $f:R\to S$ be a ring homomorphism and $J$ be an ideal of $S$. Then the subring $R\bowtie^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we will deepen the study of the prime spectrum of the amalgamations and characterize when $R\bowtie^fJ$ is a pm-ring. We also investigate the transfer of compactly packed and properly zipped property on amalgamated constructions.
Reference graph
Works this paper leans on
- [10]
-
[15]
N. Mahdou, A. Mimouni, and M. A. S. Moutui, On pm-rings, rings of finite character and h-local rings, J. Algebra Appl. 13(6), (2014), DOI: 10.1142/S0219498814500182
-
[1]
Azimi, Constructing (non-)Catenarian rings , J
Y. Azimi, Constructing (non-)Catenarian rings , J. Algebra, 597, (2022), 266–274
work page 2022
-
[2]
Y. Azimi, Hilbert rings with maximal ideals of different heights and un ruly Hilbert rings , Canadian Mathematical Bulletin, 66(1), (2022), 196–203
work page 2022
- [3]
- [4]
-
[5]
Banaschewski, pm-rings and the prime ideal theorem , Topology Appl
B. Banaschewski, pm-rings and the prime ideal theorem , Topology Appl. 158, (2011), 2340– 2342
work page 2011
-
[6]
Contessa, On some properties of pm-rings and mp-rings , C
M. Contessa, On some properties of pm-rings and mp-rings , C. R. Math. Rep. Acad. Sci. Canada 3, (1981), 357–362
work page 1981
Show all 19 references
-
[7]
Contessa, On pm-rings , Comm
M. Contessa, On pm-rings , Comm. Algebra 10, (1982), 93–108
1982
-
[8]
De Marco, A
G. De Marco, A. Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal , Proc. Amer. Math. Soc. 30, (1971), 459–466
1971
-
[9]
D’Anna, C
M. D’Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal , in: Commutative Algebra and its Applications, Proceedings of t he Fifth International Fez Con- ference on Commutative Algebra and its Applications, Fez, M orocco, 2008, W. de Gruyter Publisher,...
2008
-
[11]
D’Anna, C
M. D’Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal , Comm. Algebra, 44, (2016), 1836–1851
2016
-
[12]
D’Anna and M
M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basi c properties, J. Algebra Appl., 6, No.3, (2007), 443–459
2007
-
[13]
C. A. Finocchiaro, Pr¨ ufer-like conditions on an amalgamated algebra along an ideal, Houston J. Math., 40, (2014), 63–79
2014
-
[14]
Kaplansky, Commutative Rings , rev
I. Kaplansky, Commutative Rings , rev. ed., Univ. of Chicago Press, Chicago, (1974)
1974
-
[16]
C. J. Mulvey, A generalization of Gelfand duality , J. Algebra, 56, (1979), 499–505
1979
-
[17]
Nagata, Local Rings, Interscience, New York, (1962)
M. Nagata, Local Rings, Interscience, New York, (1962)
1962
-
[18]
C. Reis, T. Viswanathan, A compactness property for prime ideals in Noetherian rings , Proc. Amer. Math. Soc. 25, (1970), 353–356
1970
-
[19]
Tarizadeh, and J
A. Tarizadeh, and J. Chen. Avoidance and absorbance . J. Algebra, 582, (2021), 88–99. Department of Mathematics, University of Tabriz, Tabriz, Ir an. Email address : u.azimi@tabrizu.ac.ir
2021
Reviewed August 12, 2026 · model on record in the stance chip above.
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