{"id":"518fb44d-6b6c-4fa5-8714-1efded1b0aa7","arxiv_id":"1908.06744","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors introduce strongly 2-absorbing primary and strongly 2-absorbing ideals and submodules, and derive elementary properties that extend known results for strongly prime and 2-absorbing ideals.","lead":"This paper defines two new classes of ideals, strongly 2-absorbing primary ideals and strongly 2-absorbing ideals, which generalize strongly prime ideals to products of three elements. It also proves basic properties, including characterizations and intersection results, though several proof steps contain gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 2.9 uses a false ideal-closure step; the paper's main intersection result is unsupported as written, though the theorem may be repairable.","rationale":"The reader's weakest assumption identifies the same invalid step; my independent check confirms it with an explicit example in a domain where the surrounding hypotheses (nonzero strongly primary P,Q) are satisfied. The example does not refute the theorem's statement, because P and Q are comparable and the intersection is again strongly primary. It does refute the proof as written, and this matters because Theorem 2.9 is one of the two advertised structural results and is reused for submodule versions. I also examined Theorem 3.3, which the reader listed as problematical; its apparent gap (applying 2-absorbing to x,y,z that may not lie in R) can be repaired by applying 2-absorbing to xy,xz,yz∈R and then to the resulting product with 1∈R, so the conditional verdict rests on Theorem 2.9. The appropriate verdict remains CONDITIONAL, not REJECT, since the definitions, examples, and many propositions appear sound and the intersection theorem may admit a correct proof.","tokens_in":10814,"tokens_out":38948,"duration_ms":394088,"concrete_test":"Verify the displayed counterexample in the proof of Theorem 2.9: in R=F[[X^2,X^3]], set P=X^2F[[X]] and Q=X^3F[[X]], x=X^{-1}, y=X^2, z=X^2. Check that P and Q are nonzero strongly primary ideals, that xyz=X^3 lies in P∩Q, that z^1∈P and y^1∈Q, and that (xy)^1=X is not in P∩Q. This falsifies the proof's final inference. Then confirm that the theorem's conclusion still holds in this instance via (yz)^2∈P∩Q, which indicates that the concern is a proof gap requiring a corrected argument rather than a demonstrated counterexample to the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.9 is invalid at its final step. The text asserts: 'if z^n ∈ P and y^t ∈ Q, then clearly (xy)^{nt} ∈ P and (xy)^{nt} ∈ Q by definition of an ideal.' This misuses ideal closure: an ideal of R is closed under multiplication by elements of R, not by arbitrary elements of the quotient field. Concrete failure: in R=F[[X^2,X^3]], take P=X^2F[[X]] and Q=X^3F[[X]], both nonzero strongly primary ideals of R, and take x=X^{-1}, y=X^2, z=X^2. Then xyz=X^3∈P∩Q, z^1∈P, y^1∈Q, but (xy)^1=X∉P∩Q. The available information only gives ((xy)z)^n=(X^3)^n∈P and ((xy)z)^t∈Q; removing the factors z^n and y^t requires division in the quotient field, which ideals do not permit. Since Theorem 2.9 is advertised as a main result and is used in Proposition 2.12 and Proposition 3.25(c), and Theorem 3.16 is only said to be 'similar', the intersection theorems are not proved as written. The theorem may still be true, but the proof must be replaced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces two new classes of ideals: strongly 2-absorbing primary ideals and strongly 2-absorbing ideals, together with their submodule analogues. These are defined by extending the strong-prime condition to products of three elements in the quotient field of an integral domain. The paper claims that these classes generalize strongly prime ideals, and proves several basic properties: every strongly primary ideal is strongly 2-absorbing primary (Proposition 2.2), every strongly prime ideal is strongly 2-absorbing (Proposition 3.2), characterizations are given in Theorems 2.5 and 3.3, intersections of two strongly primary ideals are claimed to be strongly 2-absorbing primary (Theorem 2.9), and similarly for strongly prime ideals (Theorem 3.16). Localization, chain conditions, and minimal submodule results are also stated.","tokens_in":11104,"tokens_out":25434,"duration_ms":217360,"significance":"If the results are correct, the paper offers a reasonable common framework for strongly prime and 2-absorbing notions, and the submodule analogues are potentially useful. The definitional work is natural, and the simple implications (Propositions 2.2 and 3.2) are correct, as are several of the concrete examples. However, the main structural theorems, especially the intersection theorems and the key characterizations, rest on proofs that contain invalid steps. The paper also leaves Theorem 3.16 unproved except for a reference to the flawed proof of Theorem 2.9. The overall value of the paper will depend on whether the stated results can be supplied with correct proofs.","major_comments":[{"comment":"The proof is invalid. In the final step, the text claims that if z^n ∈ P and y^t ∈ Q, then (xy)^{nt} ∈ P and (xy)^{nt} ∈ Q 'by definition of an ideal.' This assumes that P and Q are closed under multiplication by arbitrary elements of the quotient field K, which is false since P and Q are ideals of R. Concretely, in R = F[[X^2,X^3]], take P = X^2F[[X]], Q = X^3F[[X]], x = X^{-1}, y = X^2, z = X^2. Then z ∈ P, y^2 ∈ Q, and xyz = X^3 ∈ P ∩ Q, but (xy)^2 = X^2 ∉ Q. Thus the inference is false. Since Theorem 2.9 is used in Proposition 2.12 and Proposition 3.25(c), and Theorem 3.16 is asserted to be 'similar', the intersection theorems are not established as written and require a completely new proof.","section":"§2, Theorem 2.9"},{"comment":"The proof of (b) ⇒ (a) contains an unjustified step. The text says that if xy ∈ R, xz ∈ R, and yz ∈ R, then 'we are done since I is a 2-absorbing ideal of R.' But the 2-absorbing condition applies only to triples a,b,c ∈ R, and the given x,y,z are elements of K that need not lie in R. This is not a harmless oversight: for R = Z and I = 6Z, I is 2-absorbing, and with x = 2/3, y = 3/2, z = 6 we have xyz = 6 ∈ I, while xy = 1, xz = 4, yz = 9 all lie in R and none lies in I. The proof must use condition (b) to handle this case, but no such argument is provided. This leaves the main characterization of strongly 2-absorbing ideals unproved, and all later results that invoke Theorem 3.3 are therefore conditional.","section":"§3, Theorem 3.3"},{"comment":"The proof's case analysis is not correct. After applying the strong 2-absorbing primary condition to (xy/ab)(a/x)(b/1) = y, one of the resulting alternatives is (ab/x)^n ∈ I. The proof then says 'x(ab/x) ∈ xR ⊆ I', but x(ab/x) = ab, and ab was chosen outside I. Multiplication by x ∈ I does not put ab in I unless ab/x ∈ R, which is not known. Since the argument for I^2 ⊆ J ∪ H relies on ruling out or using this case, the proof of Theorem 2.7(a) is incomplete. This also affects Corollary 2.8 and the parallel proof of Theorem 3.18.","section":"§2, Theorem 2.7(a)"},{"comment":"The proof does not justify the key implication. It states that for x ∈ K \\ R, the strong semiprime property gives x^2 ∉ P, and then 'Since P is strongly 2-absorbing, this implies that x^{-1}P ⊆ P by Theorem 3.3.' However, Theorem 3.3(b) applies to a pair x,y with xy ∉ R; taking y = x only gives x^2 ∉ P, not x^2 ∉ R, and the 'either' clause in (b) does not force x^{-1}P ⊆ P. The argument would need a different choice of the second element or a separate proof, which is not supplied. Thus the conclusion that strongly semiprime plus strongly 2-absorbing implies strongly prime is not established.","section":"§3, Proposition 3.15(a)"}],"minor_comments":[{"comment":"The sentence 'Example 3.5, Proposition 3.6, and Example 3.7 show that the converse of Proposition 2.2 is not true in general' should refer to Proposition 3.2, not Proposition 2.2, since the examples concern strongly 2-absorbing ideals and strongly prime ideals.","section":"§3, before Example 3.5"},{"comment":"The text says 'a strongly 2-absorbing prime ideal'; the paper's terminology defines 'strongly 2-absorbing ideal' and 'strongly 2-absorbing primary ideal', but not 'strongly 2-absorbing prime ideal'. This is likely a typo and should be corrected to avoid confusion.","section":"§3, Example 3.7"},{"comment":"In the proof, the line 'If x, y, and z are in V, we are done' should explicitly say that the conclusion follows from the 2-absorbing property of I, since the strong 2-absorbing condition reduces to the ordinary 2-absorbing condition when all three factors lie in V.","section":"§3, Proposition 3.11"},{"comment":"Several references are incomplete or have typographical issues: [1] gives only a page number '111' without the article's full page range, and [9] writes '441451' instead of '441–451'. These should be checked against the published versions.","section":"§1, references"},{"comment":"The notation E(S) is used before the definition of 'strongly radical' is recalled, and the connection is not immediately clear. A short explanation of the role of E(I) in the definition of strongly 2-absorbing primary would improve readability.","section":"§2, Notation 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is an early version of a definition-and-properties study. The definitions are reasonable and the simple results are correct, but several proofs of central theorems (notably Theorem 2.9, Theorem 3.3, Theorem 2.7, and Proposition 3.15) contain invalid steps or missing cases. The editor may wish to ask the authors for complete proofs of these statements before any acceptance decision; if the statements themselves are false, a more substantial revision would be needed. The paper would also benefit from a more careful comparison with existing literature on strongly prime ideals and 2-absorbing ideals, since some terminology (e.g., 'strongly semiprime') appears in slightly different forms elsewhere."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper introduces two natural ideal classes and proves a string of elementary facts about them, but the two theorems that carry the main weight are not proved as written. The definitions—strongly 2-absorbing primary ideals/submodules and strongly 2-absorbing ideals/submodules—are new relative to the cited literature and are reasonable generalizations of strongly prime ideals. Several implications are correct and easy to check: strongly primary implies strongly 2-absorbing primary (Prop. 2.2), strongly prime implies strongly 2-absorbing (Prop. 3.2), and the examples in Sections 2 and 3 are concrete, especially the K[[X^2,X^3]] examples. If all you want is the taxonomy, the first halves of Sections 2 and 3 are fine. The problem is Theorem 2.9. The proof reaches the line \"if z^n ∈ P and y^t ∈ Q, then clearly (xy)^{nt} ∈ P and (xy)^{nt} ∈ Q by definition of an ideal.\" That is not what ideal closure gives: ideals are closed under multiplication by elements of R, not by arbitrary elements of the quotient field K. Concretely, in R=F[[X^2,X^3]], take P=X^2F[[X]] and Q=X^3F[[X]], x=X^{-2}, y=X^3, z=X^2. Then xyz=X^3∈P∩Q, z∈P, y∈Q, but xy=X∉P∩Q. So the inference fails. Since Theorem 2.9 is used in Prop. 2.12 and 3.25(c), and Theorem 3.16 is only said to be \"similar,\" the intersection claims are unsupported as written. The theorem might be true and repairable, but the current proof does not establish it. I also have a smaller worry about Theorem 3.3. In the (b)⇒(a) direction, the proof says that if xy, xz, yz ∈ R then \"we are done since I is a 2-absorbing ideal of R.\" That does not follow: x,y,z are in K, and 2-absorbing is defined for triples in R. You would need an extra argument to move from (xyz)^2∈I and xy,xz,yz∈R to one of xy,xz,yz lying in I. The claim may be true, but this step is a gap. Net: the paper is worth refereeing, not desk-rejecting. The definitions and the easy propositions are fine, and the examples are real. But the two central theorems need either corrected proofs or a substantial reduction in what is claimed. I would send it to a referee who knows strongly prime ideals and ask specifically about 2.9 and 3.3.","headline":"New definitions and easy consequences are fine, but the main intersection theorem and the characterization in Theorem 3.3 are not proved as written.","tokens_in":11634,"tokens_out":10817,"would_cite":false,"duration_ms":99933,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13G05","13C13","13A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Strongly prime ideals extend to two 2-absorbing classes","keywords":["strongly prime ideal","strongly 2-absorbing primary ideal","strongly 2-absorbing primary submodule","strongly 2-absorbing ideal","strongly 2-absorbing submodule","2-absorbing ideal","rooty domain","multiplication module"],"falsifier":"Test the key step of Theorem 2.9 in a concrete non-rooty domain such as $R=K[[X^2,X^3]]$, the ring used in Example 2.3: choose nonzero strongly primary ideals $P,Q$ and elements $x,y,z$ in the quotient field with $xyz\\in P\\cap Q$, and check whether $xy\\in P\\cap Q$ or some power of $yz$ or $xz$ lies in $P\\cap Q$. If all three alternatives fail while $xyz\\in P\\cap Q$, the theorem as stated is false; if none fail, the intersection claim may survive even though its proof needs repair.","tokens_in":10621,"feed_emoji":"🧮","tokens_out":12996,"duration_ms":105543,"temperature":0.7,"pith_summary":"This paper introduces two new classes of ideals—strongly 2-absorbing primary ideals and strongly 2-absorbing ideals—as generalizations of strongly prime ideals, moving the defining \"either factor belongs to the ideal\" condition from two factors to three factors in the quotient field. The motivating claim is that the new classes sit naturally between strongly prime ideals and their ordinary 2-absorbing analogues: every strongly primary ideal is strongly 2-absorbing primary, every strongly prime ideal is strongly 2-absorbing, and in rooty domains (domains whose prime ideals are strongly radical) the new conditions admit clean characterizations in terms of inverse multiplication by quotient-field elements. The paper also proves closure properties, including intersections of strongly prime ideals and, in one of its main results, intersections of strongly primary ideals, and derives finiteness of minimal strongly 2-absorbing submodules for Noetherian modules. A sympathetic reader would care because these constructions supply a bridge between classical prime/primary decomposition and the newer 2-absorbing framework.","feed_headline":"Strongly prime ideals get 2-absorbing cousins","feed_subtitle":"Two new ideal classes generalize the quotient-field condition of strongly prime ideals to 2-absorbing settings.","key_machinery":"The load-bearing object is the quotient field $K$ of the integral domain $R$: the new definitions quantify over $x,y,z\\in K$ rather than over elements of $R$, so the ideal membership test applies even to field elements that lie outside $R$. The main working identity is the inverse-containment criterion of Theorem 3.3: a 2-absorbing ideal $I$ is strongly 2-absorbing exactly when, for every $x,y\\in K$ with $xy\\notin R$, one has $x^{-1}I\\subseteq I$ or $y^{-1}I\\subseteq I$. For the primary version, the companion mechanism is the set $E(I)$ of field elements none of whose positive powers belong to $I$, which converts \"some power enters $I$\" conclusions into a separation condition.","core_discovery":"The central discovery is that the strong-prime mechanism—checking membership of the factors of a product taken from the quotient field—can be transplanted to the 2-absorbing setting in two versions. For a 2-absorbing ideal $I$, the paper characterizes strong 2-absorbingness by a ratio-symmetric condition: for every $x,y$ in the quotient field $K$ with $xy\\notin R$, either $x^{-1}I\\subseteq I$ or $y^{-1}I\\subseteq I$ (Theorem 3.3). For strongly 2-absorbing primary ideals, it gives a parallel characterization under a rooty-domain hypothesis using the set $E(I)=\\{x\\in K:x^n\\notin I \\text{ for all } n\\ge 1\\}$ (Theorem 2.5). Along the way it proves that strongly prime ideals are strongly 2-absorbing, strongly primary ideals are strongly 2-absorbing primary, valuation domains make every 2-absorbing ideal strongly 2-absorbing, and Noetherian modules have finitely many minimal strongly 2-absorbing submodules.","pith_inferences":["Editorial inference: if Theorem 2.9 can be repaired under an added hypothesis such as rootiness or strong radicality, strongly 2-absorbing primary ideals would be closed under finite intersections, giving a 2-absorbing analogue of primary decomposition.","Editorial inference: the ratio-symmetric criterion of Theorem 3.3 suggests that strongly 2-absorbing ideals should behave well under passage to overrings, and Proposition 3.20 already shows extension to overrings; a natural next question is whether this extension is unique or canonical.","Editorial inference: because the primary version's characterization requires $K\\setminus E(I)$ to be closed under addition (true in rooty domains), one could test whether the class of strongly 2-absorbing primary ideals in non-rooty domains coincides with some valuation-theoretic condition on value groups.","Editorial inference: the finite-minimal-submodules theorem suggests a decomposition question: can every strongly 2-absorbing submodule of a Noetherian module be expressed as an intersection of minimal ones, mirroring primary decomposition?"],"forward_implications":["Every strongly prime ideal is strongly 2-absorbing, and every strongly primary ideal is strongly 2-absorbing primary, so domains that already have strongly prime ideals automatically acquire both new classes (Propositions 3.2 and 2.2).","In valuation domains, every 2-absorbing ideal is strongly 2-absorbing, so the new condition imposes no extra restriction there (Proposition 3.11).","Theorem 3.3 turns the definition into a two-variable test: to check whether a 2-absorbing ideal is strongly 2-absorbing, it suffices to test inverse containment for pairs $x,y\\in K$ with $xy\\notin R$.","The intersection of two nonzero strongly prime ideals is strongly 2-absorbing (Theorem 3.16), producing non-prime examples of strongly 2-absorbing ideals.","Every Noetherian module contains only finitely many minimal strongly 2-absorbing submodules (Theorem 3.30), so the class shares the finiteness behavior familiar from primary decomposition."],"supporting_citations":[{"why":"Defines strongly prime ideals, the base notion the paper generalizes.","marker":"[8]"},{"why":"Defines strongly primary ideals, used to seed the primary version and the intersection theorem.","marker":"[4]"},{"why":"Defines 2-absorbing ideals, the base class for strongly 2-absorbing ideals.","marker":"[3]"},{"why":"Defines 2-absorbing primary ideals, the base class for strongly 2-absorbing primary ideals.","marker":"[5]"},{"why":"Supplies the definition of rooty domains used in the characterizations of Theorems 2.5 and 2.6.","marker":"[9]"},{"why":"Supplies the fact that valuation domains are rooty, used in Proposition 3.11 and the rooty-domain hypotheses.","marker":"[2]"},{"why":"Defines strongly radical ideals, used in the $E(I)$ separation argument of Theorem 2.5.","marker":"[1]"}],"fun_headline_variants":["Strongly prime ideals spawn two 2-absorbing generalizations","From strongly prime to strongly 2-absorbing ideals","Two new ideal classes extend strongly prime condition","Strongly 2-absorbing ideals: a fresh take on primeness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise in the proof of Theorem 2.9 is that an ideal of $R$ is closed under multiplication by arbitrary elements of the quotient field: from $z^n\\in P$ and $y^t\\in Q$ the proof infers $(xy)^{nt}\\in P\\cap Q$, which requires exactly that closure, and that closure fails for a general integral domain.","fun_headline_variants_meta":{"raw":{"variants":["Strongly prime ideals spawn two 2-absorbing generalizations","From strongly prime to strongly 2-absorbing ideals","Two new ideal classes extend strongly prime condition","Strongly 2-absorbing ideals: a fresh take on primeness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000439,"raw_usage":{"total_tokens":2155,"prompt_tokens":801,"completion_tokens":1354,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":1285}},"tokens_in":417,"tokens_out":1354,"duration_ms":9636,"temperature":1.0,"reasoning_tokens":1285,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:36:48.272714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the key step of Theorem 2.9 in a concrete non-rooty domain such as $R=K[[X^2,X^3]]$, the ring used in Example 2.3: choose nonzero strongly primary ideals $P,Q$ and elements $x,y,z$ in the quotient field with $xyz\\in P\\cap Q$, and check whether $xy\\in P\\cap Q$ or some power of $yz$ or $xz$ lies in $P\\cap Q$. If all three alternatives fail while $xyz\\in P\\cap Q$, the theorem as stated is false; if none fail, the intersection claim may survive even though its proof needs repair.","supporting_citations":[{"cited_title":"Hedstrom and G","cited_arxiv_id":null,"evidence_quote":"Defines strongly prime ideals, the base notion the paper generalizes."},{"cited_title":"Badawi and E.G","cited_arxiv_id":null,"evidence_quote":"Defines strongly primary ideals, used to seed the primary version and the intersection theorem."},{"cited_title":"Badawi, On 2-absorbing ideals of commutative rings , Bull","cited_arxiv_id":null,"evidence_quote":"Defines 2-absorbing ideals, the base class for strongly 2-absorbing ideals."},{"cited_title":"Badawi, U","cited_arxiv_id":null,"evidence_quote":"Defines 2-absorbing primary ideals, the base class for strongly 2-absorbing primary ideals."},{"cited_title":"Sato and T","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of rooty domains used in the characterizations of Theorems 2.5 and 2.6."},{"cited_title":"Anderson and J","cited_arxiv_id":null,"evidence_quote":"Supplies the fact that valuation domains are rooty, used in Proposition 3.11 and the rooty-domain hypotheses."},{"cited_title":"Anderson and D.F","cited_arxiv_id":null,"evidence_quote":"Defines strongly radical ideals, used in the $E(I)$ separation argument of Theorem 2.5."}],"review_version":1}