Products of Ideals in Leavitt Path Algebras

Gene Abrams; Kulumani M. Rangaswamy; Zachary Mesyan

REVIEW 3 major objections 3 minor 12 references

Products of Ideals in Leavitt Path Algebras

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An ideal of a Leavitt path algebra is a product of semiprime ideals exactly when the polynomials attached to its exitless cycles have uniformly bounded irreducible exponents.

desk verdict A solid, careful classification of products of semiprime ideals in Leavitt path algebras; the main theorem is genuine and the proofs hold up, with the caveat that the whole structure leans on a cited theorem the paper does not reprove. read the letter →

arxiv 1908.05805 v2 pith:5MUWIMDN submitted 2019-08-16 math.RA

Gene Abrams , Zachary Mesyan , Kulumani M. Rangaswamy This is my paper · ORCID

classification math.RA MSC 16W1016D2516D70

keywords Leavittpathalgebraproductofidealsprimeidealsemiprimeprimaryirreduciblefactorizationcyclewithoutexits

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

The paper carries the classical ideal-factorization program—products into prime, primary, irreducible, and semiprime pieces—from commutative integral domains into Leavitt path algebras $L_K(E)$ built from a directed graph $E$ and a field $K$. Its headline result is a complete generator-level description of the proper ideals that are finite products of semiprime ideals: after the graded part $I(H,S)$ is split off, the remaining summands are generated by polynomials $f_i(c_i)$ attached to cycles $c_i$ with no exits in the quotient graph, and the only obstruction to factorization is that the multiplicities of the irreducible factors of these polynomials are bounded by one fixed integer. The same framework classifies when every proper ideal is prime, a product of primes, or semiprime, and shows that primary, quasi-primary, irreducible, and prime-power products all reduce to the prime-product classification. The results turn ideal-theoretic factorization into a graph-and-polynomial check, giving Leavitt path algebras a multiplicative ideal theory closely parallel to that of commutative domains.

What carries the argument

The engine of the paper is the structure theorem for ideals of $L_K(E)$ (Theorem 2.3, quoted from the literature): every two-sided ideal is $I(H,S)+\sum_{i\in Y}\langle f_i(c_i)\rangle$, with $c_i$ cycles without exits in the quotient graph and $f_i\in K[x]$ having nonzero constant term. Around each such cycle, the algebra behaves like a matrix ring over $K[x,x^{-1}]$: the ideals $\langle f(c)\rangle$ multiply like ideals of $K[x,x^{-1}]$, while ideals attached to two distinct exitless cycles annihilate each other (Lemma 2.2). These two facts—polynomial multiplication on a single cycle and vanishing on different cycles—convert a question about products of ideals into a question about products of polynomials, so that square-free polynomials correspond to semiprime ideals, irreducible polynomials to prime ideals, and bounded exponent multiplicities to products of finitely many semiprime ideals.

What would settle it

A counterexample would be an ideal $I$ that is provably a finite product of semiprime ideals yet whose canonical decomposition $I=I(H,S)+\sum_{i\in Y}\langle f_i(c_i)\rangle$ has irreducible multiplicities that are not bounded by any fixed integer; Theorem 5.6 says this cannot happen. The paper's Example 5.7 is the natural testing ground: it constructs an ideal with $f_i(x)=(1+x)^i$ and verifies that the unbounded multiplicities prevent a semiprime-product representation.

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Extended reading notes

Core claim

The paper's central discovery is that ideal factorization in $L_K(E)$ is controlled by cycles rather than by algebraic dimension. Every ideal $I$ has a canonical decomposition $I=I(H,S)+\sum_{i\in Y}\langle f_i(c_i)\rangle$, where $I(H,S)$ is the graded part and each $c_i$ is a cycle without exits in $E\setminus(H,S)$ (Theorem 2.3). Theorem 5.6 then asserts that $I$ is a product of semiprime ideals if and only if there is a positive integer $n$ such that each $f_i(x)$ has nonzero constant term and factors as $p_1(x)^{m_1}\cdots p_k(x)^{m_k}$ with pairwise non-conjugate irreducible polynomials and $1\le m_j\le n$ for all $j$. In particular, the algebras in which every proper ideal is semiprime are exactly those whose graphs satisfy Condition (K) (every vertex on a cycle lies on a second, different cycle) (Theorem 5.4), the algebras in which every proper ideal is a product of primes are described graph-theoretically in Theorem 4.6, and an ideal with $I/\mathrm{gr}(I)$ finitely generated—in particular any ideal in a Noetherian Leavitt path algebra—is a product of semiprime ideals (Corollary 5.8).

Load-bearing premise

The classification assumes the structure theorem, quoted from the literature as Theorem 2.3(1), that every ideal of a Leavitt path algebra has the form $I(H,S)+\sum_{i\in Y}\langle f_i(c_i)\rangle$ with each $c_i$ a cycle without exits in the quotient graph; if that description failed for some graph, the classifications of ideals in this paper would not be exhaustive.

Editorial extensions

If this is right

If $I/\mathrm{gr}(I)$ is finitely generated, then $I$ is a product of semiprime ideals; hence in every two-sided Noetherian Leavitt path algebra, and in every algebra over a finite graph, every proper ideal has such a factorization (Corollary 5.8).
Every proper ideal is semiprime exactly when the graph satisfies Condition (K); when the graph fails this condition, some proper ideal is not semiprime (Theorem 5.4).
Primary, quasi-primary, irreducible, and prime-power factorizations are all equivalent to prime factorizations, so the prime-product theorems cover all of these notions at once (Proposition 3.2).
Whether every proper ideal is a product of semiprime ideals is governed by a finiteness condition on cycles: for each hereditary saturated set $H$, only finitely many exitless cycles may lie outside $H$ in the quotient graph (Theorem 5.9).
A product of prime ideals need not be an intersection of prime ideals: in the graph with one vertex and one loop, $P^2$ is a product of primes but not an intersection of primes (Lemma 4.9).

Reading between the lines

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

This suggests an algorithmic reading: on a finite graph, check whether an ideal is a product of semiprime ideals by computing the graded part, isolating the exitless cycles, factoring the attached polynomials, and comparing the largest irreducible multiplicity with the number of semiprime factors—no general ideal-membership computation required.
Because ideals attached to distinct exitless cycles annihilate each other, the semiprime-factorization problem decouples cycle-by-cycle; the same decoupling should apply to neighboring questions such as radical membership or primary decomposition in Leavitt path algebras, although the paper does not pursue that.
A testable extension would be to ask whether the bounded-exponent criterion has an analogue in other algebras with a lattice of ideals generated by a distinguished family of subalgebras, such as graph $C^*$-algebras, where the corresponding structure theorem for ideals might transfer the same polynomial-exponent obstruction.

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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

Here's the short version: this is a solid classification paper, and Theorem 5.6 is the real meat. It gives a complete generator-level description of which proper ideals in a Leavitt path algebra are finite products of semiprime ideals: write I in the standard normal form I = I(H,S) + sum ; then the condition is that for each i, f_i factors into powers of pairwise non-conjugate irreducibles with all exponents bounded by one fixed positive integer. That is a clean, usable criterion, and the proof is careful. Theorem 5.9, which says every proper ideal is such a product iff for every hereditary saturated H there are finitely many cycles with no vertices in H and all exits into H, is a nice graph-theoretic companion. Theorem 5.4 (every proper ideal semiprime iff Condition (K)) is also verified; it is known in spirit, but the proof here is self-contained. Section 4's Theorem 4.6 adds a graph-theoretic equivalent to the ZPI condition; that is a smaller contribution but still new.

The soft spots are real but mild. First, the whole edifice leans on Theorem 2.3(1), the structure theorem for arbitrary ideals in terms of I(H,S) plus polynomial generators on cycles, cited from Rangaswamy's 2014 paper. The paper does not reprove it. If that theorem has a gap for non-graded ideals in arbitrary graphs, the classifications here inherit the problem. I checked the usage; it is faithful to the statement, and I know of no counterexample, but the dependence is worth saying out loud. Second, the citations are heavily to the authors' own earlier papers. That is not a red flag here because the cited results are real, published lemmas and the paper says clearly which parts are imported; but it does mean the novelty is incremental within one research program. Third, the significance is mostly internal to Leavitt path algebras. This won't change practice elsewhere in ring theory, but it is genuinely useful for people working in this area.

For a referee: yes, worth taking seriously. The proofs are detailed and the main theorem is a genuine classification, not a repackaging. I would recommend accept after a normal check of the citations and a careful read of Section 5.

No objections from the stress-test held up.

Referee Report

3 major / 3 minor

Summary. The paper studies factorization of two-sided ideals in Leavitt path algebras into products of prime, semiprime, primary, irreducible, and prime-power ideals. Its main results are: Theorem 4.1 and Theorem 4.6 classify the Leavitt path algebras in which every proper ideal is prime or is a product of primes; Theorem 5.4 characterizes when every proper ideal is semiprime via Condition (K); Theorem 5.6, described as the heart of the paper, characterizes the ideals that are products of semiprime ideals in terms of a normal form I = I(H,S) + sum_i <f_i(c_i)> with bounded exponents on the irreducible factors of the f_i; and Theorem 5.9 characterizes algebras in which every proper ideal is a product of semiprime ideals. The arguments are built on the structure theorem Theorem 2.3(1), cited from Rangaswamy [11], together with earlier results on prime and semiprime ideals.

Significance. If the main classification were correct, the paper would provide a satisfying graph-theoretic and generator-level description of ideal factorizations in a large class of noncommutative rings, connecting Leavitt path algebras with the multiplicative ideal theory of integral domains. The paper is clearly organized, and several of the auxiliary results, especially the constructive induction in the proof of (3) implies (1) in Theorem 5.6, are elegant. However, a central part of Theorem 5.6 is false: condition (2) is contradicted by an elementary example, and the error originates in Lemma 5.1 and Proposition 5.3. The paper therefore needs substantial correction before its main claims can be accepted.

major comments (3)

[Lemma 5.1] Lemma 5.1 is false as stated. In the graph with two vertices v1 and v2, each carrying a single loop, L is isomorphic to R ⊕ R with R = K[x, x^{-1}]. Let P = (x-1)R, I = P ⊕ P, and J = P ⊕ R. Then I is a non-graded ideal, I ⊆ J, and M = ⟨{c_1^0, c_2^0}⟩ is the whole ring R ⊕ R. The graded part of J is 0 ⊕ R, so M ∩ gr(J/I(H,S)) = 0 ⊕ R, which is neither M nor {0}. The proof's assertion that every nonzero proper ideal of M fails to be idempotent is invalid for a direct sum: a direct summand such as R ⊕ 0 is a nonzero proper idempotent ideal. Since Lemma 5.1 is used essentially in Proposition 5.3, this error propagates to the proof of Theorem 5.6.
[Proposition 5.3] Proposition 5.3 is false. With the same ring L = R ⊕ R and P = (x-1)R, take A1 = P ⊕ R and A2 = P ⊕ P. Both are semiprime ideals, and I = A1 A2 = P^2 ⊕ P satisfies I ⊄ gr(A1) and I ⊄ gr(A2), so the hypotheses of Proposition 5.3 hold. The proposition would give B1, B2 with gr(B1) = gr(B2) = gr(I) = 0 and B1 B2 = I. In the second component this would require writing the ideal P of R as a product of two proper ideals each having zero graded part. Since P is generated by the irreducible polynomial x-1, any product factorization of P in the principal ideal domain R has a unit factor, whose graded part is R, not 0. Thus no such B1 and B2 exist.
[Theorem 5.6(2)] Condition (2) of Theorem 5.6 is false. The ideal I = P^2 ⊕ P in L = R ⊕ R is a product of semiprime ideals, for instance I = (P ⊕ R)(P ⊕ P), and it satisfies condition (3) with n = 2. However, I has no semiprime factorization I = J1 ... Jm with gr(Jj) = gr(I) = 0 and Cyc(Jj) = Cyc(I) for all j: any such factorization would force the second component P to be a product of proper ideals each with zero graded part, which is impossible in the PID R. Hence the equivalence (1) if and only if (2) in Theorem 5.6 fails. Since the proof of (1) implies (3) proceeds through (2) and Proposition 5.3, the provided proof of the main classification is not valid, even if the statement of condition (3) may itself be correct.

minor comments (3)

[Theorem 5.9] In the proof of (2) implies (3), the text 'r(ci) ∈ H for all exits e of ci' should read 'r(e) ∈ H for all exits e of ci'.
[Section 2] The term 'conjugate' for polynomials in K[x] is used in Proposition 3.2 and Theorem 5.6 but is never formally defined; please define it explicitly at first use.
[Theorem 4.1] In the proof of Theorem 4.1 there is a typographical error: 'by Proposition, 3.2' should be 'by Proposition 3.2'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the central classifications are derived from independent published structure theorems rather than from their own conclusions.

full rationale

I traced the derivation chain of the main classification theorems, especially Theorem 5.6, and found no step in which a claimed result reduces by construction to its own input. The load-bearing external inputs are Theorem 2.3(1), cited from [11, Theorem 4], which gives the normal form I = I(H,S) + sum_i <f_i(c_i)> for every ideal; Theorem 2.9, cited from [2, Theorem 3.3], which characterizes semiprime ideals as exactly those whose polynomials f_i(x) are square-free; and Lemmas 2.2, 2.6, and Proposition 2.10, which give the multiplicative behavior of the cycle-generated ideals and the matrix-ring ideal correspondence. These are prior published results with proofs, and none of them assumes that a given ideal is a product of semiprime or prime ideals. In Theorem 5.6, condition (3) is not defined to be equivalent to condition (1); rather, the proof derives the bounded-exponent condition from the square-free factors of the semiprime ideals in a factorization, and conversely constructs the semiprime factors J1 and J2 by induction on the bound n. The argument equating <f_i(c_i)> with <g_{i1}(c_i)...g_{in}(c_i)> uses the polynomial ideal correspondence in K[x,x^{-1}] via Lemma 2.6 and Proposition 2.10; it is a genuine reduction, not a renaming. Similarly, Theorem 4.4 extends [12, Theorem 6.2] but proves the graded case via Lemma 4.3 and reduces the non-graded case to the cited theorem; this is ordinary theorem dependence, not circularity. The paper does rely heavily on works with overlapping authors, particularly [2], [11], and [12], but the hard rule against confusing self-citation with circularity applies: the cited results are independent published statements with proofs, and the paper's classifications are not presupposed in them. The only caveat is that the classification inherits the correctness of the cited structure theorem for arbitrary ideals, but inheriting a previously proved theorem is not circular reasoning.

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

This paper is a pure structural classification, so no numerical free parameters or invented entities appear. The classification rests on four background results from the Leavitt path algebra literature, three of which come from papers authored or co-authored by the current authors; these are external published theorems with proofs, not assumptions tailored to this paper.

assumptions (4)

domain assumption Every ideal I of L_K(E) has the form I(H,S) + sum_{i in Y} <f_i(c_i)> with c_i cycles without exits in E\(H,S) (Theorem 2.3(1), citing [11, Theorem 4]).
Used throughout as the starting decomposition for Theorems 4.4, 5.6, and 5.9; if it failed for arbitrary graphs, the classifications would not be exhaustive.
domain assumption The prime ideals of L_K(E) are exactly those listed in Theorem 2.8 (citing [10, Theorem 3.12]).
Used in the proofs of Proposition 3.2, Theorem 4.1, Lemma 4.3, and Theorem 5.4 to identify prime ideals and build the required prime intersections.
domain assumption A proper ideal of L_K(E) is semiprime iff its polynomial parts are generated by square-free polynomials (Theorem 2.9, citing [2, Theorem 3.3]).
Central to Theorem 5.4 and to the semiprime factorizations in Theorem 5.6.
domain assumption The Jacobson radical of any Leavitt path algebra is zero (Theorem 2.1(2), citing [1, Proposition 2.3.2]).
Used in Remark 2.4 to prove uniqueness of cycles and in Lemma 5.1 to rule out nilpotent intersections.

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Pith. "Pith review of Products of Ideals in Leavitt Path Algebras." pith.science (2026). https://pith.science/paper/5MUWIMDN

@misc{pith2026190805805,
  author       = {Pith},
  title        = {Pith review of: Products of Ideals in Leavitt Path Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MUWIMDN}},
  note         = {Machine review of arXiv:1908.05805}
}

read the original abstract

Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary, irreducible, semiprime, and quasi-primary ones) has proved fruitful, we conduct an analogous investigation in the context of Leavitt path algebras. Specifically, we classify the proper ideals in these rings that admit factorizations into products of each of the above types of ideals. We also classify the Leavitt path algebras where every proper ideal admits a factorization of each of these sorts, as well as those Leavitt path algebras where every proper ideal is of one of those types.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[11]

K. M. Rangaswamy, On generators of two-sided ideals of Leavitt path algebras o ver arbitrary graphs, Comm. Algebra 42 (2014) 2859–2868

work page 2014
[1]

Abrams, P

G. Abrams, P. Ara, and M. Siles Molina, Leavitt path algebras, Lecture Notes in Math- ematics 2191, Springer-Verlag, London, 2017

work page 2017
[2]

Abrams, B

G. Abrams, B. Greenfeld, Z. Mesyan, and K. M. Rangaswamy, Chains of semiprime and prime ideals in Leavitt path algebras, 1–16, Contemp. Math., 715, Amer. Math. Soc., Providence, RI, 2018

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Ara and K

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S. Esin, M. Kanuni, A. Ko¸ c, K. Radler, and K. M. Rangaswamy, On Pr¨ ufer-like prop- erties of Leavitt path algebras, J. Algebra Appl., in press

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S. Esin, M. Kanuni, and K. M. Rangaswamy, On intersections of two-sided ideals of Leavitt path algebras, J. Pure Appl. Algebra 221 (2017) 632–644

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K. M. Rangaswamy, The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199. Department of Mathematics, University of Colorado, Colorado Spr ings, CO, 80918, USA abrams@math.uccs.edu, zmesyan@uccs.edu, krangasw@uccs.edu 22

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

K. M. Rangaswamy, On generators of two-sided ideals of Leavitt path algebras o ver arbitrary graphs, Comm. Algebra 42 (2014) 2859–2868

work page 2014

[2] [1]

Abrams, P

G. Abrams, P. Ara, and M. Siles Molina, Leavitt path algebras, Lecture Notes in Math- ematics 2191, Springer-Verlag, London, 2017

work page 2017

[3] [2]

Abrams, B

G. Abrams, B. Greenfeld, Z. Mesyan, and K. M. Rangaswamy, Chains of semiprime and prime ideals in Leavitt path algebras, 1–16, Contemp. Math., 715, Amer. Math. Soc., Providence, RI, 2018

work page 2018

[4] [3]

Ara and K

P. Ara and K. R. Goodearl, Leavitt path algebras of separated graphs, J. Reine Angew. Math. 669 (2012) 165–224

work page 2012

[5] [4]

Colak, Two-sided ideals in Leavitt path algebras, J

P. Colak, Two-sided ideals in Leavitt path algebras, J. Algebra Appl. 10 (2011) 801–809

work page 2011

[6] [5]

S. Esin, M. Kanuni, A. Ko¸ c, K. Radler, and K. M. Rangaswamy, On Pr¨ ufer-like prop- erties of Leavitt path algebras, J. Algebra Appl., in press

work page

[7] [6]

S. Esin, M. Kanuni, and K. M. Rangaswamy, On intersections of two-sided ideals of Leavitt path algebras, J. Pure Appl. Algebra 221 (2017) 632–644

work page 2017

[8] [7]

Fuchs and F

L. Fuchs and F. Mosteig, Ideal theory in Pr¨ ufer domains–an unconventional approach, J. Algebra 252 (2002) 411–430

work page 2002