{"id":"456f1428-bf39-4689-aa39-4c45bd7842c1","arxiv_id":"2506.02989","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new notion, (u,v)-absorbing primary hyperideals, is introduced for multiplicative hyperrings, with structural results and examples, several of which need correction.","lead":"This paper defines a new type of ideal called a (u,v)-absorbing primary hyperideal inside multiplicative hyperrings, a setting where multiplying two numbers can yield a set of results. The authors prove several properties of these ideals and give examples, though some examples and proof steps contain errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.7's proof relies on an unjustified inference from a∘b⊆rad(P) to a^n∘b^n⊆P, and this step underpins the central claim that radicals of (u,v)-absorbing primary C-hyperideals are prime.","rationale":"The reader assigned CONDITIONAL, citing among other issues a logical gap in Theorem 2.7's handling of units and an unproved radical equality in Proposition 2.8. My stress-test identifies a different, more severe gap in the same theorem: the unjustified inference from a∘b⊆rad(P) to a^n∘b^n⊆P. This step is load-bearing because Theorem 2.7 is used to prove several later results, including the Dedekind hyperdomain characterization (Theorem 2.18) and the intersection theorem (Proposition 2.15). I do not claim the theorem is false; it may be repairable by adding a Noetherian/finiteness hypothesis or by a different proof. However, the central claim as stated is not established. This does not change the reader's verdict: the paper still needs substantial correction before acceptance, and the new concept is plausibly sound. I therefore keep CONDITIONAL (via UNCHANGED) rather than moving to REJECT, since no counterexample to the main definition is known and several results appear structurally correct. The agreement is partial because the reader focused on a different weak point in the same proof rather than the uniform-power inference.","tokens_in":16429,"tokens_out":21621,"duration_ms":216284,"concrete_test":"Test the ordinary-commutative-ring analogue of Theorem 2.7, where every ideal is a C-hyperideal and hyperproducts are singletons. Determine whether every ideal I of a (not necessarily Noetherian) commutative ring satisfying x1...xu∈I for nonunits x1,...,xu implies x1...xv∈I or x_{v+1}...x_u∈rad(I) has prime radical. A counterexample for u=3,v=1 or u=3,v=2 would refute Theorem 2.7; a proof that rad(I) is prime would show the theorem is plausible but that the paper's uniform-power inference still needs replacement. Alternatively, in a multiplicative hyperring with infinite hyperoperations (e.g., Z_Φ with Φ infinite), search for a (u,v)-absorbing primary C-hyperideal P and elements a,b with a∘b⊆rad(P) but a,b∉rad(P); such a pair would directly invalidate the proof's key step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 2.7 (Section 2), after assuming a∘b⊆rad(P) for a,b∈A\\U(A), the author states: 'Then we get a^n∘b^n⊆P for some n∈N.' This assertion is not proved and does not follow from the definitions. From a∘b⊆rad(P) one only knows that each element of the hyperproduct a∘b has some power in P, with the exponent depending on that element. The C-hyperideal property gives rad(P) = {x | x^m⊆P for some m} (via Proposition 3.2 of [13]), but it does not bound these exponents. Without a finiteness or Noetherian-type hypothesis, there is no reason that a single n works for every element of a^n∘b^n, which contains products of n elements chosen from a∘b. The subsequent application of Definition 2.1 to x1=...=xv−1=a, x_v=x, x_{v+1}=...=xu−1=b, x_u=y requires exactly this unjustified inclusion. Since Theorem 2.7 is used in the proofs of Theorem 2.14, Proposition 2.15, Theorem 2.18 and Theorem 2.19, the central structural assertion that rad(P) is prime for every (u,v)-absorbing primary C-hyperideal is not established as written. The missing step is not a minor cosmetic gap: without a valid proof of Theorem 2.7, the claimed hierarchy placing this class between primary and (u,v)-absorbing prime hyperideals rests on an unproven foundation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces (u,v)-absorbing primary hyperideals in commutative multiplicative hyperrings, a notion intended to interpolate between primary hyperideals and the author's earlier (u,v)-absorbing prime hyperideals. The main results are: the radical of a (u,v)-absorbing primary C-hyperideal is prime (Theorem 2.7); a sufficient condition for P∘M to be such a hyperideal in a local hyperring (Proposition 2.8); a locality criterion when a stronger absorbing property is not inherited (Theorem 2.12); intersection and colon properties (Propositions 2.9 and 2.15); characterizations in Dedekind hyperdomains (Theorem 2.18) and divided multiplicative hyperrings (Theorem 2.19); and stability under homomorphisms, quotients, localizations, and matrix hyperrings (Theorems 2.20-2.24). The paper also presents examples separating the new class from (u,v)-absorbing prime hyperideals.","tokens_in":16762,"tokens_out":15940,"duration_ms":148367,"significance":"If the main structural claims are correct, the definition is a natural extension of the ring-theoretic (u,v)-absorbing primary ideals to multiplicative hyperrings, and the paper provides a useful collection of tools (intersections, colons, residues, localization, and matrix constructions) for working with this class. The paper's strengths are its clean definition, its explicit use of the C-hyperideal and strong C-hyperideal machinery, and its systematic treatment of standard hyperring constructions. However, the current version contains several load-bearing gaps: the proof of Theorem 2.7 relies on an unjustified uniform-exponent step, Proposition 2.8 invokes an unproved radical equality, Example 2.16 contains an arithmetic error, and Theorem 2.18 omits the zero-ideal case and an unproved maximality assertion. These issues affect the paper's central hierarchy claim and must be repaired before the results can be regarded as sound. There is no computational verification; the examples are hand-checked and several have notational or arithmetic problems.","major_comments":[{"comment":"The proof of Theorem 2.7 contains an unjustified step: from a∘b⊆rad(P) the author concludes 'a^n∘b^n⊆P for some n∈N'. For a C-hyperideal, rad(P) is the set of elements some power of which lies in P, but this provides one exponent per element of a∘b and no uniform bound on those exponents. Moreover, a^n∘b^n consists of products of n elements chosen from a and n from b, not powers of single elements of a∘b, so the inclusion does not follow from the definition. The subsequent application of Definition 2.1 with x1=...=xv−1=a, x_v=x, x_{v+1}=...=x_{u−1}=b, and x_u=y depends entirely on this step. Since Theorem 2.7 is used in the proofs of Theorem 2.14, Proposition 2.15, Theorem 2.18, and Theorem 2.19, this gap threatens the paper's central structural assertion. The proof needs either a correct argument producing a uniform exponent, or an additional hypothesis (for example, a Noetherian-like finiteness condition).","section":"Section 2, Theorem 2.7"},{"comment":"The proof of Proposition 2.8 uses the equality P = rad(P∘M) without proof, where P is a prime hyperideal and M is the unique maximal hyperideal. In classical commutative rings this equality can be shown when P⊆M, but the hyperring analogue is not automatic and the paper provides neither a proof nor a citation. If the equality fails, the step 'xv+1∘...∘xu⊆P=rad(P∘M)' collapses, and the claim that P∘M is (u,v)-absorbing primary is unsupported. Please either prove the radical equality from the hyperring axioms or state explicitly the hypotheses under which it holds.","section":"Section 2, Proposition 2.8"},{"comment":"Example 2.16 asserts ⟨150⟩=⟨3⟩∩⟨5⟩∩⟨7⟩ in the multiplicative hyperring Z_Φ with Φ={2,4}. In the underlying ring Z, the intersection of these ideals is ⟨105⟩ (150 is not divisible by 7), so the example is arithmetically incorrect. Since the example is meant to show that the equal-radical condition in Proposition 2.15 is crucial, it does not support that claim as written. In addition, the hyperring Z_Φ with Φ={2,4} does not have an identity element in the sense of the paper (an element e with a∈e∘a for every a), because no integer e satisfies 1∈{2e,4e}; this conflicts with the standing assumption that A has identity element 1. Similar concerns affect other examples using Φ not containing 1. The example should be recalculated or replaced.","section":"Section 2, Example 2.16"},{"comment":"The reverse implication in Theorem 2.18 uses the assertion 'Since rad(P) is prime, by the hypothesis, rad(P) is maximal.' The paper does not prove that every nonzero prime hyperideal in a Dedekind hyperdomain is maximal, and the statement is false for P=0, where rad(P)=0 is not maximal unless the hyperdomain is a hyperfield. The later step ⟨y, rad(P)⟩=A for y∉rad(P) depends on maximality. Please supply the missing argument for Dedekind hyperdomains or explicitly exclude the zero ideal and justify the maximality claim.","section":"Section 2, Theorem 2.18"}],"minor_comments":[{"comment":"The title contains the typo 'MUL TIPLICA TIVE', and the introduction contains 'hyerrings' and 'hperrings'; these should be corrected.","section":"Title/Abstract"},{"comment":"The introduction states u,v∈Z, while the rest of the paper uses u,v∈N with u>v; the definition should be stated consistently with N.","section":"Introduction"},{"comment":"The first sentence of Section 2 contains the typo 'A proper hyperideal ofPofAis said to be...' and should read 'A proper hyperideal P of A is said to be...'.","section":"Section 2, first paragraph"},{"comment":"In Example 2.2(i), the description of P=2Z[i] as a set of the form {-2x-2yi, 6x+6yi : x,y∈Z} is confusing and likely misprinted; the usual description is {2a+2bi : a,b∈Z}. In Example 2.2(ii), α and β are said to be in Z, but the hyperring is A=Z+3xZ[x], so the hyperoperation should be defined for elements of A.","section":"Example 2.2"},{"comment":"The displayed computation of 2^3∘3 in Example 2.4 does not match the hyperoperation with Φ={2,3}; please recalculate the set of products.","section":"Example 2.4"},{"comment":"In Example 2.6, the expression '2◦2 = {8,12}/∈P' should read '2◦2 = {8,12}⊈P'.","section":"Example 2.6"},{"comment":"In Theorem 2.20(i), η^{-1}(P2) is said to be a C-hyperideal of A2; it should be of A1. In part (ii), the text contains 'P1 is a is a' and says η(P1) is a C-hyperideal of A1; the latter should be A2.","section":"Theorem 2.20"},{"comment":"In the proof of Theorem 2.23, the notation '⊈S^{-1}Q' should be '⊈S^{-1}P'.","section":"Theorem 2.23"},{"comment":"The conclusion contains the typo 'it ,s radical', and Section 4 (Future work) contains a new Definition 4.1; including a formal definition in a future-work section is surprising and should be either moved to the main development or removed.","section":"Conclusion and Section 4"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own recent work [5,6], and the citation pattern should be checked after revision. More importantly, the examples need systematic re-verification against the stated hyperring axioms: several use Z_Φ with Φ not containing a multiplicative identity, which conflicts with the paper's standing assumption 'with identity element 1'. The arithmetic error in Example 2.16 and the unjustified step in Theorem 2.7 are the main obstacles to accepting the paper in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces (u,v)-absorbing primary hyperideals in commutative multiplicative hyperrings. That is a legitimate new object, sitting naturally between the author's earlier (u,v)-absorbing prime hyperideals and the ring-theoretic (u,v)-absorbing primary ideals. The paper does some useful orienting work: it shows every (u,v)-absorbing prime is (u,v)-absorbing primary, gives a non-prime example, and states stability properties under intersections, localization, and homomorphisms. Several of those proofs are checkable and fine, and the definition is precise enough to be falsifiable. Credit is due for that.\n\nThe soft spots are real and not cosmetic. Theorem 2.7, which claims rad(P) is prime for a (u,v)-absorbing primary C-hyperideal, has a logical gap. From a∘b⊆rad(P) the author jumps to a^n∘b^n⊆P for a single n. That does not follow: each element of a∘b may have its own exponent, and without a finiteness or Noetherian-type hypothesis there is no uniform bound. The stress-test note is right about this. Since Theorem 2.14, 2.15, 2.18, and 2.19 all lean on Theorem 2.7, the main structural hierarchy is not established as written. The result may be true with extra hypotheses, but the paper does not supply them.\n\nThe other clear error is Example 2.16, which asserts <150> = <3>∩<5>∩<7> in Z. The intersection is <105>. That example is supposed to show the radical-equality condition in Proposition 2.15 is crucial, and as written it fails. This is a trivial fix, but it indicates the manuscript was not checked carefully. Proposition 2.8 also uses rad(P∘M)=P without proof, which needs hypotheses; the reader flagged this and the concern is fair.\n\nThe self-citations are heavy but not circular. The new definition explicitly builds on prior work, and the results are checkable from the stated definitions and standard facts, not from a hidden reuse of the target claim.\n\nNet: the definition is worth having in the subfield, but the paper is not in a reliable state. It needs a repaired Theorem 2.7 (or a clearly stated extra hypothesis), a corrected Example 2.16, and a proof or explicit condition for the radical equality in Proposition 2.8. I would still send it to a serious referee—the topic is narrow but legitimate, and the results are plausibly repairable. I would not cite it in my own work until the main theorem is fixed.","headline":"A legitimate new definition in a niche subfield, but the central structural theorem has a real proof gap and one example is arithmetically wrong; worth refereeing, not worth trusting as is.","tokens_in":744,"tokens_out":863,"would_cite":false,"duration_ms":30093,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20N20","16Y20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines $(u,v)$-absorbing primary hyperideals in commutative multiplicative hyperrings, proves that their radical is prime for $C$-hyperideals, and characterizes the class in Dedekind and divided hyperdomains.","keywords":["multiplicative hyperring","(u,v)-absorbing primary hyperideal","primary hyperideal","prime radical","C-hyperideal","hyperdomain","local hyperring"],"falsifier":"Compute $\\operatorname{rad}(P\\circ M)$ in a concrete local multiplicative hyperring such as the $\\mathbb{Z}_\\Phi$ construction with $\\Phi=\\{2,3\\}$, taking $P$ a prime hyperideal and $M$ the unique maximal hyperideal; if the radical is strictly larger than $P$, or if $P\\circ M$ fails the $(u,v)$-absorbing primary condition, Proposition 2.8 is false. For Theorem 2.18, look for a Dedekind hyperdomain with a $C$-hyperideal whose radical is prime but which is not primary, which would refute the claimed equivalence.","tokens_in":16209,"feed_emoji":"🧮","tokens_out":11764,"duration_ms":101386,"temperature":0.7,"pith_summary":"Commutative multiplicative hyperrings replace ordinary multiplication by a set-valued hyperoperation, and this paper transplants the ring notion of $(u,v)$-absorbing primary ideals into them. A proper hyperideal $P$ is $(u,v)$-absorbing primary when a product of $u$ non-units contained in $P$ forces either the first $v$ factors into $P$ or the remaining factors into the radical $\\operatorname{rad}(P)$. The paper aims to show this class sits strictly between $(u,v)$-absorbing prime hyperideals and primary hyperideals: every prime version is a primary version, the converse fails by example, and for $C$-hyperideals the radical of the new object is always prime. It then proves two characterization results: in a Dedekind hyperdomain a $C$-hyperideal is $(u,v)$-absorbing primary exactly when its radical is prime, and in a divided hyperring the new class is just the primary hyperideals. A further structural theorem says that a strong $C$-hyperideal that is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary without being $(u,v)$-absorbing primary can exist only when the whole hyperring is local.","feed_headline":"Radical of a (u,v)-absorbing primary hyperideal is prime","feed_subtitle":"For a C-hyperideal the new class has prime radical; in Dedekind hyperdomains that condition is exact.","key_machinery":"The load-bearing mechanism is the $C$-hyperideal condition together with the prime radical. A $C$-hyperideal is one for which a finite hyperproduct that merely meets the hyperideal is forced to lie entirely inside it, and a strong $C$-hyperideal does the same for finite sums of hyperproducts; this is what lets the proofs pass from a single containment to radical membership and back. The radical tracks the primary half of the definition, and Theorem 2.7 uses the $C$-condition to show that $\\operatorname{rad}(P)$ inherits primality. In the Dedekind hyperdomain result, invertibility of hyperideals supplies maximality of the prime radical; in the locality theorem, the strong $C$-property on $\\operatorname{rad}(P)$ is used to deduce that the sum of a non-unit and a unit is a unit, which is the criterion for a local hyperring.","core_discovery":"The central claim is that Definition 2.1 gives a useful intermediate class in commutative multiplicative hyperrings. For integers $u>v$, a proper hyperideal $P$ is $(u,v)$-absorbing primary if $x_1\\circ\\cdots\\circ x_u\\subseteq P$ with $x_1,\\ldots,x_u\\notin U(A)$ implies $x_1\\circ\\cdots\\circ x_v\\subseteq P$ or $x_{v+1}\\circ\\cdots\\circ x_u\\subseteq \\operatorname{rad}(P)$. The paper establishes that every $(u,v)$-absorbing prime hyperideal is $(u,v)$-absorbing primary and that the reverse fails (Example 2.4), and that the primary property is monotone in both parameters. Its main structural assertions are Theorem 2.7, that the radical of a $(u,v)$-absorbing primary $C$-hyperideal is prime; Theorem 2.12, that a strong $C$-hyperideal which is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary but not $(u,v)$-absorbing primary forces the hyperring to be local; and Theorem 2.18, that in a Dedekind hyperdomain a $C$-hyperideal is $(u,v)$-absorbing primary if and only if its radical is prime. Proposition 2.8 further claims that in a local hyperring whose hyperideals are $C$-hyperideals, the product $P\\circ M$ of a prime hyperideal with the maximal hyperideal is $(u,v)$-absorbing primary.","pith_inferences":["As an extension beyond the paper, Definition 4.1's $(u,v)$-absorbing $I$-primary variant could be tested against the Section 2 theorems; replacing $x_1\\circ\\cdots\\circ x_u\\subseteq P$ by $x_1\\circ\\cdots\\circ x_u\\subseteq P\\setminus IP$ would show how much of the hierarchy survives.","A natural converse question the paper does not ask is whether every local multiplicative hyperring contains a $(u+1,v+1)$- or $(u+1,v)$-absorbing primary strong $C$-hyperideal that is not $(u,v)$-absorbing primary; Theorem 2.12 only proves the one-way implication.","The scope of Proposition 2.8 hinges on the unproved equality $\\operatorname{rad}(P\\circ M)=P$; proving or disproving that equality for arbitrary local multiplicative hyperrings would decide whether the product construction is as broad as stated.","If the new class is genuinely distinct only outside divided hyperrings, a concrete test is to check whether the $\\mathbb{Z}_\\Phi$ construction with $\\Phi=\\{2,3\\}$ has $(u,v)$-absorbing primary hyperideals whose radicals are prime but which are not primary."],"forward_implications":["Every $(u,v)$-absorbing prime hyperideal is automatically $(u,v)$-absorbing primary, and Example 2.4 shows the converse fails, so the primary version is strictly broader.","For a $(u,v)$-absorbing primary $C$-hyperideal, the radical is a prime hyperideal (Theorem 2.7), matching the radical behaviour of classical primary ideals.","In a Dedekind hyperdomain, a $C$-hyperideal is $(u,v)$-absorbing primary if and only if its radical is prime (Theorem 2.18).","In a divided multiplicative hyperring, the $(u,v)$-absorbing primary $C$-hyperideals are exactly the primary hyperideals (Theorem 2.19).","A strong $C$-hyperideal that is $(u+1,v+1)$- or $(u+1,v)$-absorbing primary but not $(u,v)$-absorbing primary can exist only in a local multiplicative hyperring (Theorem 2.12)."],"supporting_citations":[{"why":"Supplies the ring-level $(m,n)$-absorbing primary ideals that Definition 2.1 generalizes to multiplicative hyperrings.","marker":"[23]"},{"why":"Defines $(u,v)$-absorbing prime hyperideals; the paper's primary version replaces the second containment by radical membership, and Example 2.4 compares the two classes.","marker":"[6]"},{"why":"Defines 1-absorbing primary hyperideals, the base case that the new two-parameter definition extends.","marker":"[5]"},{"why":"Supplies the definitions of prime and primary hyperideals, the prime radical, and the $C$-hyperideal properties used in Theorems 2.7, 2.15, and 2.17.","marker":"[13]"},{"why":"Provides the Dedekind hyperdomain definition, the locality criterion used in Theorem 2.12, and background on divided hyperrings used in Theorem 2.19.","marker":"[19]"},{"why":"Supplies foundational terminology for multiplicative hyperrings: units, maximal hyperideals, local hyperrings, hyperideal quotients, and hypermatrix rings.","marker":"[2]"},{"why":"Supplies the behaviour of $C$-hyperideals under good homomorphisms used to prove Theorem 2.20.","marker":"[31]"},{"why":"Supplies the localization of hyperideals invoked in Theorems 2.23 and 2.24.","marker":"[26]"}],"fun_headline_variants":["Radical of (u,v)-absorbing primary C-hyperideal is prime","For C-hyperideals, (u,v)-absorbing primary implies prime radical","In Dedekind hyperdomains, (u,v)-absorbing primary iff prime radical","Prime radical characterizes (u,v)-absorbing primary C-hyperideals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that inside a local multiplicative hyperring, the radical of the product $P\\circ M$ of a prime hyperideal $P$ with the maximal hyperideal $M$ is exactly $P$; the proof of Proposition 2.8 uses this equality without proving it, and the proposition collapses if it is false.","fun_headline_variants_meta":{"raw":{"variants":["Radical of (u,v)-absorbing primary C-hyperideal is prime","For C-hyperideals, (u,v)-absorbing primary implies prime radical","In Dedekind hyperdomains, (u,v)-absorbing primary iff prime radical","Prime radical characterizes (u,v)-absorbing primary C-hyperideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000464,"raw_usage":{"total_tokens":2287,"prompt_tokens":880,"completion_tokens":1407,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":1318}},"tokens_in":496,"tokens_out":1407,"duration_ms":11355,"temperature":1.0,"reasoning_tokens":1318,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:13:21.459913+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\operatorname{rad}(P\\circ M)$ in a concrete local multiplicative hyperring such as the $\\mathbb{Z}_\\Phi$ construction with $\\Phi=\\{2,3\\}$, taking $P$ a prime hyperideal and $M$ the unique maximal hyperideal; if the radical is strictly larger than $P$, or if $P\\circ M$ fails the $(u,v)$-absorbing primary condition, Proposition 2.8 is false. For Theorem 2.18, look for a Dedekind hyperdomain with a $C$-hyperideal whose radical is prime but which is not primary, which would refute the claimed equivalence.","supporting_citations":[{"cited_title":"Khali, H","cited_arxiv_id":null,"evidence_quote":"Supplies the ring-level $(m,n)$-absorbing primary ideals that Definition 2.1 generalizes to multiplicative hyperrings."},{"cited_title":"(u,v)-absorbing (prime) hyperideals in commutative multiplicative hyperrings","cited_arxiv_id":"2406.14276","evidence_quote":"Defines $(u,v)$-absorbing prime hyperideals; the paper's primary version replaces the second containment by radical membership, and Example 2.4 compares the two classes."},{"cited_title":"Anbarloei, On 1-absorbing prime hyperideal and some of its generalizations,Journal of Mathematics, (2022)","cited_arxiv_id":null,"evidence_quote":"Defines 1-absorbing primary hyperideals, the base case that the new two-parameter definition extends."},{"cited_title":"Dasgupta, On prime and primary hyperideals of a multiplicative hyperring,Annals of the Alexandru Ioan Cuza University-Mathematics,L VIII (1) (2012) 19-36","cited_arxiv_id":null,"evidence_quote":"Supplies the definitions of prime and primary hyperideals, the prime radical, and the $C$-hyperideal properties used in Theorems 2.7, 2.15, and 2.17."},{"cited_title":"Ghiasvand, F","cited_arxiv_id":null,"evidence_quote":"Provides the Dedekind hyperdomain definition, the locality criterion used in Theorem 2.12, and background on divided hyperrings used in Theorem 2.19."},{"cited_title":"Ameri, A","cited_arxiv_id":null,"evidence_quote":"Supplies foundational terminology for multiplicative hyperrings: units, maximal hyperideals, local hyperrings, hyperideal quotients, and hypermatrix rings."},{"cited_title":"Sengelen Sevim, B.A","cited_arxiv_id":null,"evidence_quote":"Supplies the behaviour of $C$-hyperideals under good homomorphisms used to prove Theorem 2.20."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the localization of hyperideals invoked in Theorems 2.23 and 2.24."}],"review_version":1}