{"id":"6571574a-6807-470f-9882-cc222abf983c","arxiv_id":"1908.04880","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Serre-type category equivalence is claimed for semi-graded rings under a noetherian and underexplained X1 condition, and a degenerate class of semi-graded Artin-Schelter algebras is identified.","lead":"This mathematics paper reworks five famous open problems in noncommutative algebra for a broad class of 'skew PBW' rings and proves a Serre-type equivalence for these rings under certain conditions. A generalist might care because such rings include many quantum algebras used in mathematical physics, and the paper's framework offers a common language for attacking the problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.24 rests on an undefined hypothesis X1 and its proof applies Proposition 1.23 to the auxiliary graded ring B-script rather than to the original semi-graded ring B, so the stated qgr-B equivalence is not established.","rationale":"I agree with the reader that X1 being undefined is a serious problem. But the more decisive structural issue is that the proof of Theorem 1.24 proves an equivalence for the auxiliary graded ring B-script, not for the original B, unless an unstated identification is supplied. Original B is only semi-graded: BmBn is contained in the sum B0⊕...⊕Bm+n, not in Bm+n, so Proposition 1.23 cannot be applied with C = B. The natural source for ρ is the N-graded ring B-script defined in Lemma 1.22(ii). Its quotient category qgr-B-script is not the same object as qsgr-B for arbitrary B, and the paper gives no functor between them beyond Γ. In the finitely graded case Corollary 1.25(iii) proves B ≅ B-script, which explains why the classical Serre equivalence works, but for general semi-graded rings this identification is absent. The paper does contain useful independent material: the Maple-implemented Quillen-Suslin algorithm in §3.1 is concrete and reproducible, and the SPBWE free resolutions in §4 give computational support for the SAS examples. Those results are not affected by the gap in Theorem 1.24. Because the theorem's hypothesis and its source category are both under-specified, the correct verdict is UNVERDICTED rather than conditional acceptance.","tokens_in":27170,"tokens_out":16928,"duration_ms":174824,"concrete_test":"Locate the definition of X1 (presumably in [37], where Lemma 6.10 lives) and check whether it coincides with the χ1 condition of §1.1(vii). Then rewrite the proof of Theorem 1.24 with the source ring C in Proposition 1.23 specified explicitly: if C is the ring B-script of Lemma 1.22(ii), supply a proof or reference for an equivalence qgr-B ≃ qgr-B-script for semi-graded B satisfying (C1)-(C4) and the stated X1. If no definition of X1 and no such equivalence can be found, Theorem 1.24 is unsupported as written.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Theorem 1.24 is the paper's central new claim, so its proof must connect a well-defined hypothesis to the stated conclusion. It does not. First, 'B satisfies X1' is never defined: §1.1(vii) defines χi and χ, but X1 appears only in Lemma 1.22(v), Theorem 1.24, Corollary 1.25, and Example 1.26. Lemma 1.22(v) invokes Proposition 3.1.3(3) of Artin-Zhang [10], and the proof of Theorem 1.24 invokes part S10 of Theorem 4.5 of [10], on the strength of X1. If X1 is not exactly the χ1 condition, or if χ1 alone does not imply the needed finite generation of Γ0 and Γd and the right-boundedness of kernel and cokernel, the argument has no footing. Second, Proposition 1.23 requires its source C to be an N-graded noetherian algebra. The original semi-graded B is not N-graded in general; the only N-graded ring available in Lemma 1.22 is B-script := ⊕ Hom_{sgr-B}(B, s^d(B)). The proof sets C to that auxiliary ring, so the category equivalence it yields is qgr-B-script ≃ qgr-Γ, not qgr-B ≃ qgr-Γ. The identification qgr-B ≃ qgr-B-script (or qsgr-B ≃ qgr-B-script) is neither proved nor cited. Thus Theorem 1.24's stated conclusion does not follow even if X1 is later defined.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of five open problems in noncommutative algebra and noncommutative projective algebraic geometry, formulated in the language of skew PBW extensions and semi-graded rings. It reviews the Serre-Artin-Zhang-Verevkin theorem, the Gelfand-Kirillov conjecture, Serre's problem on projective modules, Artin-Schelter regularity, and the Zariski cancellation problem. The new results claimed are Theorem 1.24 (an equivalence qgr-B ≃ qgr-Γ(π(B))≥0 for semi-graded rings satisfying (C1)-(C4) and a condition called X1), Corollary 1.25 (applications to associated graded rings and to finitely graded algebras), and Theorem 4.14 (a class of bijective skew PBW extensions whose members are semi-graded Artin-Schelter regular algebras). The paper also contains numerous explicit free resolutions, computed with the SPBWE library, for quantum algebras and other examples in Section 4.","tokens_in":27479,"tokens_out":6744,"duration_ms":67686,"significance":"If Theorem 1.24 were fully established, it would remove the domain assumption from the earlier result in [37] and extend the Serre-Artin-Zhang-Verevkin equivalence to semi-graded rings; that would be a substantive contribution. The concrete free resolutions in Section 4 are useful computational evidence and the reformulation of several open problems in a common framework could be valuable. However, the proof of Theorem 1.24 has load-bearing gaps: the condition X1 is never defined, the category qgr-B is not defined for a merely semi-graded ring, and the proof works with an auxiliary N-graded ring rather than with B itself. The SAS notion in Section 4 is also weak because many examples satisfy the key homological condition only vacuously. The paper is therefore not yet self-contained or conclusive on its central claim, although the identified problems appear repairable.","major_comments":[{"comment":"The hypothesis X1 is never defined. The text defines χ_i and χ in §1.1(vii), but X1 appears without definition in Lemma 1.22(v), Theorem 1.24, Corollary 1.25, and Example 1.26. The proof of Lemma 1.22(v) invokes Proposition 3.1.3(3) of [10] and the proof of Theorem 1.24 invokes part S10 of Theorem 4.5 of [10] on the strength of X1. If X1 is not exactly the χ_1 condition, or if χ_1 alone does not imply the finite generation of Γ_d and the right boundedness of the kernel and cokernel of ρ, then Lemma 1.22(v) and Theorem 1.24 are not established. This is load-bearing because Theorem 1.24 is advertised as the main new result of the paper.","section":"§1.1(vii), Lemma 1.22(v), Theorem 1.24"},{"comment":"The proof of Theorem 1.24 does not establish the stated equivalence qgr-B ≃ qgr-Γ(π(B))≥0. Proposition 1.23 requires its source algebra to be N-graded. The original semi-graded ring B is not N-graded in general; the only N-graded ring available in Lemma 1.22 is the auxiliary ring B-script := ⊕_d Hom_{sgr-B}(B, s^d(B)). The proof applies Proposition 1.23 to the homomorphism from this auxiliary ring, and would therefore yield an equivalence qgr-B-script ≃ qgr-Γ(π(B))≥0. No argument is given to identify qgr-B with qgr-B-script, or qsgr-B with qgr-B-script. Moreover, the notation qgr-B in the statement is itself undefined for a merely semi-graded ring: §1.1(v) defines qgr only for finitely graded algebras, while for semi-graded rings the paper constructs the category qsgr-B in §1.3. The conclusion of Theorem 1.24 is therefore not derived as written.","section":"Theorem 1.24 and Proposition 1.23"},{"comment":"The definition of B≥s as 'the least two-sided ideal of B' satisfying conditions (a) and (b) asserts existence and uniqueness without proof. The torsion submodule T(M), the Serre subcategory stor-B, and the quotient category qsgr-B all depend on this ideal, so the construction on which Theorem 1.24 rests is not fully justified. The paper should either prove existence and uniqueness of the least ideal or replace the definition by an explicit construction, for example as the intersection of all two-sided ideals with the stated properties.","section":"Definition 1.15"},{"comment":"For several of the displayed SAS examples, condition (iv) of Definition 4.3 is satisfied only because B≥1 = B and therefore B/B≥1 = 0; see Examples 4.5, 4.6, 4.7 and 4.12, and the proof of Theorem 4.14. This makes the regularity condition vacuous in these cases. The definition is formally coherent, but these examples do not provide independent evidence that SAS regularity is a strong homological property, and the significance of Theorem 4.14 is accordingly weaker than the surrounding text suggests.","section":"Definition 4.3 and Examples 4.5-4.7, 4.12, 4.14"}],"minor_comments":[{"comment":"The algorithm loops 'FOR k from 1 to n−1', but Theorem 3.1 and Example 3.3 concern a single variable x; the role of n in the algorithm should be clarified.","section":"§3.1, Algorithm 1"},{"comment":"The manuscript contains numerous typographical errors, including 'satifies' in Theorem 4.14(v), 'conmutative' in Definition 5.1, and 'Adv. Mah.' in reference [10]; a careful proofreading pass is needed.","section":"Global"},{"comment":"The isomorphism B-script ≅ B for finitely graded B is asserted with the comment 'As (i), we can prove that θ is an isomorphism'; the surjectivity argument should be written out, since it depends on the finitely graded hypothesis in a way that is not immediate from the preceding parts.","section":"Corollary 1.25(iii)"},{"comment":"References [20] and [40] are cited as 'to appear' without preprint identifiers; please update them if final versions or arXiv numbers are available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely a survey with a few new results. The central new theorem is not self-contained and the proof has a category-mismatch issue, but the gaps are of a kind that could be repaired by adding definitions and a comparison argument between qgr and qsgr. I recommend major revision rather than rejection, and the journal may wish to consider whether the survey material is appropriately balanced with the new results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this for the survey and the computational examples, not for Theorem 1.24. The useful core is the reformulation of five open problems—Serre-type equivalence, Gelfand-Kirillov, Quillen-Suslin, AS regularity/noetherianity, Zariski cancellation—in the language of skew PBW extensions and semi-graded rings, together with a good deal of concrete data: free resolutions for the dispin algebra, U(sl(2)), U(so(3)), U'(so(3)), the Woronowicz algebra, and several families of 3-dimensional skew polynomial algebras. Those computations are real, and the survey is a competent entry point to the semi-graded point of view.\n\nThe advertised new theorems are another matter. Theorem 1.24 asserts qgr-B ≃ qgr-Γ(π(B))≥0 for semi-graded B satisfying (C1)–(C4) and a hypothesis printed as \"X1\". X1 is never defined. I assume it is a typo for χ1, but as written the theorem's key hypothesis is unusable. More seriously, the proof of Theorem 1.24 applies Proposition 1.23 with C = B, but Proposition 1.23 requires an N-graded noetherian algebra. The original B is only semi-graded. The N-graded object in Lemma 1.22 is the auxiliary ring B-script = ⊕ Hom_{sgr-B}(B, s^d(B)); the map ρ in the proof is defined on elements of that ring, so what the argument actually yields is qgr-B-script ≃ qgr-Γ, not qgr-B ≃ qgr-Γ. No equivalence between qgr-B and qgr-B-script is proved or cited. This is a gap in the central claim, not a typo. Lemma 1.22(v) also imports boundedness conclusions from Artin-Zhang [10] without checking the hypotheses, so the proof is not self-contained.\n\nTheorem 4.14 is much weaker than the introduction suggests. Condition (v) forces A/A≥1 = 0, so the SAS Ext condition in Definition 4.3 is satisfied vacuously; the noetherian-domain conclusion promised in the introduction is not actually part of the theorem and is only gestured at in a remark. The examples around it are still worth having.\n\nBottom line: this deserves a serious referee because the survey and the examples are useful and the Serre-type theorem may be repairable, but Theorem 1.24 needs a genuinely fixed proof and Theorem 4.14 needs honest reframing. I would not cite Theorem 1.24 as it stands. I would still send the paper to peer review rather than desk-reject: there is enough solid content here that a careful referee could help the author turn it into a reliable survey with a corrected main theorem.","headline":"The paper is a useful survey with real computational examples, but its advertised new Serre-type theorem (Theorem 1.24) is not proved as written: the argument runs through an auxiliary graded ring and the key hypothesis X1 is never properly defined.","tokens_in":28079,"tokens_out":7690,"would_cite":false,"duration_ms":78499,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16S36","16U20","16D40","16E05","16E65","16S38","16S80","16W70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Semi-graded rings satisfy the Serre-Artin-Zhang-Verevkin equivalence without a domain assumption, provided they meet the X1 finiteness condition, and a broad class of skew PBW extensions are noetherian domains in the semi-graded…","keywords":["semi-graded rings","skew PBW extensions","noncommutative projective schemes","Serre-Artin-Zhang-Verevkin theorem","Artin-Schelter regular algebras","Gelfand-Kirillov conjecture","Zariski cancellation problem","Quillen-Suslin theorem"],"falsifier":"Compute the kernel and cokernel of the canonical ring map from the graded endomorphism ring of $B$ to $\\Gamma(\\pi(B))_{\\ge 0}$ for a left noetherian semi-graded ring that satisfies (C1)–(C4) and X1 but is not a domain—for instance a matrix ring over a skew PBW extension. If either is not right-bounded, or if some homogeneous component of a finitely generated $\\Gamma$-module over $\\Gamma_0$ is not finitely generated, then Proposition 1.23 cannot be applied and the equivalence of Theorem 1.24 fails.","tokens_in":26867,"feed_emoji":"🔁","tokens_out":12954,"duration_ms":113431,"temperature":0.7,"pith_summary":"This paper reformulates five open problems of noncommutative algebra and noncommutative geometry—the Serre-type equivalence, the Gelfand-Kirillov conjecture, Serre's freeness problem, the noetherianity and domain questions for Artin-Schelter regular algebras, and Zariski cancellation—in the language of skew PBW extensions and semi-graded rings. Its main new result removes the domain restriction from an earlier version of the Serre-Artin-Zhang-Verevkin theorem: for a left noetherian semi-graded ring satisfying conditions (C1)–(C4) and a finiteness condition X1, the quotient category $\\mathrm{qgr}\\text{-}B$ is equivalent to $\\mathrm{qgr}\\text{-}\\Gamma(\\pi(B))_{\\ge 0}$. A second new theorem shows that bijective skew PBW extensions satisfying mild homogeneity and nonvanishing hypotheses are semi-graded Artin-Schelter regular algebras, placing examples like the Weyl algebra, the dispin algebra, and quantum algebras inside the theory. If these results stand, noncommutative projective geometry extends to rings that are only semi-graded rather than finitely graded, and the evidence that Artin-Schelter regular algebras are noetherian domains broadens.","feed_headline":"Semi-graded rings get Serre's equivalence without the domain axiom","feed_subtitle":"The quotient-category equivalence holds under the X1 finiteness condition—no domain assumption needed.","key_machinery":"The machinery is the pair of categories built from a semi-graded ring $B$: $\\mathrm{sgr}\\text{-}B$, the abelian category of finitely generated semi-graded modules, and its quotient $\\mathrm{qsgr}\\text{-}B=\\mathrm{sgr}\\text{-}B/\\mathrm{stor}\\text{-}B$ by the Serre subcategory of torsion modules, where torsion is defined by annihilation by powers of the least two-sided ideal $B_{\\ge s}$ containing the high-degree part. The bridge to the graded world is the functor $\\Gamma(\\pi(B))_{\\ge 0}=\\bigoplus_{d\\ge 0}\\mathrm{Hom}_{\\mathrm{qsgr}\\text{-}B}(\\pi(B),s^d\\pi(B))$ that packages the shifts $s$ of the structure sheaf; the equivalence is obtained by applying a graded Morita-type comparison (Proposition 1.23) to the ring map $B^\\# \\to \\Gamma(\\pi(B))_{\\ge 0}$, whose kernel and cokernel are shown to be right-bounded using the X1 condition. The condition X1 is thus the load-bearing import from the theory of noncommutative projective schemes, making the graded ring $\\Gamma$ left noetherian with finitely generated homogeneous components.","core_discovery":"On the paper's own terms, the central discovery is that the Serre-Artin-Zhang-Verevkin equivalence is a semi-graded phenomenon, not a graded one. Theorem 1.24 states that if $B=\\bigoplus_{n\\ge 0}B_n$ is a left noetherian semi-graded ring satisfying (C1)–(C4)—$B_0$ left noetherian, each $B_n$ finitely generated over $B_0$, and $B_0$ central—and satisfies X1, then $\\mathrm{qgr}\\text{-}B \\simeq \\mathrm{qgr}\\text{-}\\Gamma(\\pi(B))_{\\ge 0}$, where $\\pi(B)$ is the image of $B$ in the quotient of finitely generated semi-graded modules by torsion and $\\Gamma$ collects the shifts of $\\pi(B)$. Corollary 1.25 draws out the consequences: the associated graded ring $\\mathrm{Gr}(B)$ recovers the same category when it is left noetherian and satisfies X1, and for finitely graded $B$ the old equivalence is recovered. The paper also proves (Theorem 4.14) that a bijective skew PBW extension over a connected, finite-semi-graded, finite-left-global-dimension base, with homogeneous endomorphisms and derivations and at least one nonzero constant $d_{ij}$, is a semi-graded Artin-Schelter regular algebra; the worked examples include the Weyl algebra, the $q$-Weyl algebra, the dispin algebra, the enveloping algebras of $\\mathfrak{sl}(2,K)$ and $\\mathfrak{so}(3,K)$, and the Woronowicz algebra, with five of the eight 3-dimensional skew polynomial algebra types being SAS and three not.","pith_inferences":["If X1 coincides with the standard $\\chi_1$ finiteness condition, as the argument appears to intend, then Theorem 1.24 suggests that the right hypothesis for a Serre-style theory is not grading but the combination of noetherianity and $\\chi$-finiteness; this could be tested by checking whether semi-graded rings with zero divisors but with $\\chi_1$, such as certain matrix or monoid algebras, admit t","The reliance on the least ideal $B_{\\ge s}$ in the definition of torsion is the most delicate point for non-domains; a natural test is whether this ideal exists and equals the ordinary irrelevant ideal in every bijective skew PBW extension, or only under the stated noetherian-plus-central hypotheses.","The SAS framework could be used to test Zariski cancellation beyond graded Artin-Schelter algebras: the noetherian SAS algebras of global dimension 3 exhibited here are natural candidates for the Makar-Limanov or centralizer criteria, and a counterexample among them would sharpen the boundary of the cancellation property."],"forward_implications":["Theorem 1.24 gives the Serre-Artin-Zhang-Verevkin equivalence for all semi-graded rings satisfying (C1)–(C4) and X1, not only for domains as in the earlier treatment.","Corollary 1.25 links the quotient category of a semi-graded ring to that of its associated graded ring: when $\\mathrm{Gr}(B)$ is left noetherian and satisfies X1, $\\mathrm{qgr}\\text{-}\\mathrm{Gr}(B)\\simeq\\mathrm{qgr}\\text{-}B$.","The known semi-graded examples—enveloping algebras of Lie algebras, $\\mathrm{U}'(\\mathfrak{so}(3,K))$, dispin, Woronowicz, and the eight 3-dimensional skew polynomial types—all satisfy the hypotheses, so the equivalence holds for them.","Theorem 4.14 supplies semi-graded Artin-Schelter regular algebras that are noetherian domains, and the examples separate SAS algebras from essentially regular ones: the Weyl and $q$-Weyl algebras are SAS, while $K\\{x,y\\}/\\langle yx-xy+y\\rangle$ and $R_{yx}$ are not.","If $\\mathrm{Gr}(B)$ is left noetherian and satisfies X1, condition X1 transfers back to $B$, so semi-graded rings with nice associated graded rings enter the same noncommutative projective geometry."],"supporting_citations":[{"why":"Supplies the finiteness condition X1 and the boundedness results (Proposition 3.1.3(3) and Theorem 4.5) that the proof of Lemma 1.22(v) and Theorem 1.24 imports to make $\\Gamma(\\pi(B))_{\\ge 0}$ noetherian with right-bounded kernel and cokernel.","marker":"[10]"},{"why":"Introduces semi-graded rings, the categories $\\mathrm{sgr}\\text{-}B$ and $\\mathrm{stor}\\text{-}B$, and proves the Serre-type equivalence under a domain restriction that Theorem 1.24 removes.","marker":"[37]"},{"why":"Defines skew PBW extensions, the main class of examples whose semi-graded structure motivates the problems and Theorem 1.24.","marker":"[36]"},{"why":"Provides the homological facts (finite left global dimension, Ore domain properties) used in Theorem 4.14 to show certain skew PBW extensions are semi-graded Artin-Schelter regular algebras.","marker":"[38]"},{"why":"Introduces Artin-Schelter regular algebras, the objects whose open noetherianity and domain questions are reformulated for semi-graded rings.","marker":"[9]"},{"why":"Gives the Ore and Goldie localization results for skew PBW extensions that underlie the Gelfand-Kirillov discussion in Problem 2.","marker":"[2]"},{"why":"Defines the cancellation notions and the center/Makar-Limanov criteria that support the Zariski cancellation problem for SAS algebras.","marker":"[13]"}],"fun_headline_variants":["Serre equivalence holds for semi-graded rings without the domain axiom","Semi-graded rings satisfy Serre's equivalence under X1","Serre-Artin-Zhang equivalence extended to semi-graded rings","Domain axiom dropped for Serre's equivalence in semi-graded setting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a technical finiteness condition named X1 that the paper cites from the graded theory but never defines; the main equivalence goes through only if X1 really has the boundedness and finiteness consequences the proof needs.","fun_headline_variants_meta":{"raw":{"variants":["Serre equivalence holds for semi-graded rings without the domain axiom","Semi-graded rings satisfy Serre's equivalence under X1","Serre-Artin-Zhang equivalence extended to semi-graded rings","Domain axiom dropped for Serre's equivalence in semi-graded setting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001071,"raw_usage":{"total_tokens":4530,"prompt_tokens":1031,"completion_tokens":3499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":3423}},"tokens_in":647,"tokens_out":3499,"duration_ms":24557,"temperature":1.0,"reasoning_tokens":3423,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:30:32.156088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the kernel and cokernel of the canonical ring map from the graded endomorphism ring of $B$ to $\\Gamma(\\pi(B))_{\\ge 0}$ for a left noetherian semi-graded ring that satisfies (C1)–(C4) and X1 but is not a domain—for instance a matrix ring over a skew PBW extension. If either is not right-bounded, or if some homogeneous component of a finitely generated $\\Gamma$-module over $\\Gamma_0$ is not finitely generated, then Proposition 1.23 cannot be applied and the equivalence of Theorem 1.24 fails.","supporting_citations":[{"cited_title":"and Zhang J","cited_arxiv_id":null,"evidence_quote":"Supplies the finiteness condition X1 and the boundedness results (Proposition 3.1.3(3) and Theorem 4.5) that the proof of Lemma 1.22(v) and Theorem 1.24 imports to make $\\Gamma(\\pi(B))_{\\ge 0}$ noetherian with right-bounded kernel and cokernel."},{"cited_title":"and Latorre, E","cited_arxiv_id":null,"evidence_quote":"Introduces semi-graded rings, the categories $\\mathrm{sgr}\\text{-}B$ and $\\mathrm{stor}\\text{-}B$, and proves the Serre-type equivalence under a domain restriction that Theorem 1.24 removes."},{"cited_title":"and Gallego, C","cited_arxiv_id":null,"evidence_quote":"Defines skew PBW extensions, the main class of examples whose semi-graded structure motivates the problems and Theorem 1.24."},{"cited_title":"& Reyes, M","cited_arxiv_id":null,"evidence_quote":"Provides the homological facts (finite left global dimension, Ore domain properties) used in Theorem 4.14 to show certain skew PBW extensions are semi-graded Artin-Schelter regular algebras."},{"cited_title":", Graded algebras of global dimension 3 , Advances in Mathematics, 66, 1987, 171-216","cited_arxiv_id":null,"evidence_quote":"Introduces Artin-Schelter regular algebras, the objects whose open noetherianity and domain questions are reformulated for semi-graded rings."},{"cited_title":", Ore and Goldie theorems for skew PBW extensions , Asian-European Journal of Mathematics, 6 (4), 2013, 1350061 - 1; 1350061-20","cited_arxiv_id":null,"evidence_quote":"Gives the Ore and Goldie localization results for skew PBW extensions that underlie the Gelfand-Kirillov discussion in Problem 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the cancellation notions and the center/Makar-Limanov criteria that support the Zariski cancellation problem for SAS algebras."}],"review_version":1}