A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

Dawei Shen

REVIEW 2 major objections 3 minor 25 references

A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proves that n-Gorenstein rings are detected by when indecomposable injectives first appear versus their flat dimension, then shows that in Auslander-Gorenstein Nakayama algebras every odd Ext group of a simple module is either zer

desk verdict A genuinely useful Serre-type criterion and a strong Nakayama application; the main claims look right, but the Artin-algebra transfer and one iterated-syzygy step need tightening. read the letter →

arxiv 2607.17497 v2 pith:N2R4WCDN submitted 2026-07-20 math.RT

Dawei Shen This is my paper · ORCID

classification math.RT MSC 16E1016G20

keywords n-GorensteinringsAuslander-GorensteinSerre-typecriterionNakayamaalgebrassyzygyfiltrationoccurrencedegreeflatdimensiongradeofsimplemodules

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's central claim is a Serre-type recognition theorem: a two-sided Noetherian ring is n-Gorenstein—meaning the first n terms of a minimal injective coresolution have flat dimension bounded by their degree—exactly when every indecomposable injective module appears in that coresolution no earlier than the smaller of n and its flat dimension. For Artin algebras this becomes a comparison between the grade of each simple module and the projective dimension of its injective envelope. The paper then applies the criterion to Nakayama algebras (Artin algebras whose indecomposable modules are uniserial) through syzygy filtration, proving that n-Gorensteinness reduces to 2-Gorensteinness plus (n−2)-Gorensteinness of a smaller algebra ε(A). Iterating, the Auslander-Gorenstein property is preserved and reflected, and selfinjective dimension drops by two. The payoff is structural: for an Auslander-Gorenstein Nakayama algebra, odd Ext groups Ext^n(S,A) are nonzero exactly when the projective dimension of S is n, and in that case they are simple right modules—so simple modules of odd grade are regular and odd terms of the minimal injective coresolution are multiplicity-free.

What carries the argument

The carrying mechanism is the syzygy-filtration category. For a Nakayama algebra A, one selects base modules ∇(S) attached to simple modules S with pdim S ≠ 1; these modules form a semibrick, and their filtration category E is an exact abelian subcategory of mod A that is equivalent to the module category of the syzygy-filtered algebra ε(A) = End_A^op(⊕ P), itself a Nakayama algebra of smaller rank. The identity doing the work is that the embedding E → mod A preserves projective covers, projective dimension, and all Ext groups: pdim_E X = pdim_A X and Ext^n_E ≅ Ext^n_A. Since the second syzygy of any module lies in E, an n-degree Ext computation over A becomes an (n−2)-degree computation ove

What would settle it

Compute the odd-degree terms of the minimal injective coresolution of a small Auslander-Gorenstein Nakayama algebra (for instance from a Kupisch series): if any indecomposable injective appears in an odd degree with multiplicity greater than one, or appears in a degree not equal to the projective dimension of its simple module, then Theorem 5.1 and its multiplicity-free corollary are false.

Watch

Extended reading notes

Core claim

The paper's central claim is an if-and-only-if criterion: a two-sided Noetherian ring R is n-Gorenstein precisely when occ_R(I) ≥ min{n, fdim_R I} for every indecomposable injective R-module I, where occ_R(I) is the first degree at which I appears in a minimal injective coresolution of R and fdim_R I its flat dimension. For Artin algebras this becomes: grade_A S ≥ min{n, pdim_A I(S)} for every simple S, with equality when grade_A S < n. The Nakayama application is a reduction formula: for n ≥ 2, A is n-Gorenstein iff A is 2-Gorenstein and ε(A) is (n−2)-Gorenstein, where ε(A) is the syzygy-filtered algebra. Hence A is Auslander-Gorenstein iff it is 2-Gorenstein and ε(A) is Auslander-Gorenstei

Load-bearing premise

The whole Nakayama argument rests on the assumption that the syzygy-filtration machinery from the author's earlier paper, originally proved for finite-dimensional algebras over an algebraically closed field, transfers unchanged to basic Artin Nakayama algebras satisfying (N1)–(N2); the only given justification is that the proofs rely solely on uniseriality and on mutual isomorphism of the endomorphism rings of simple modules.

Editorial extensions

If this is right

n-Gorensteinness of any two-sided Noetherian ring is equivalent to a check on first-occurrence degrees and flat dimensions of indecomposable injectives; no injective coresolution needs to be written out.
For Nakayama algebras, n-Gorensteinness is equivalent to 2-Gorensteinness of every iterated syzygy-filtered algebra up to ⌊n/2⌋, giving a finite reduction procedure.
The Auslander-Gorenstein property is preserved by syzygy filtration and, among 2-Gorenstein Nakayama algebras, is reflected by it; when it holds, the selfinjective dimension drops by two each step.
In an Auslander-Gorenstein Nakayama algebra, an odd Ext group Ext^n_A(S,A) vanishes unless pdim_A S = n, and when nonzero it is a simple right module; simple modules of odd grade are therefore regular.
The odd-degree terms of the minimal injective coresolution of an Auslander-Gorenstein Nakayama algebra are multiplicity-free: each injective envelope of a simple appears in degree n exactly when pdim S = n, and with multiplicity one.

Reading between the lines

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

The inequality occ_R(I) ≤ fdim_R I, proven here for module-finite algebras over a commutative Noetherian ring, is left open for arbitrary two-sided Noetherian rings; if it held generally, the (G_n) criterion would give a fully general Serre-type characterization of n-Gorenstein rings.
The reduction formula suggests a practical algorithm: from a Kupisch series, iterate ε until a selfinjective algebra is reached; checking 2-Gorenstein at each step would decide Auslander-Gorensteinness without computing injective resolutions.
The odd/even asymmetry in Theorem 5.1 leaves a natural open direction: test whether even-degree Ext groups or even-degree injective-coresolution terms satisfy any bounded-multiplicity or simplicity statement under stronger hypotheses, for instance higher selfinjective dimension.
Because the proof of Theorem 5.1(2) works by showing the relevant higher transpose is simple, one could ask whether the conjunction of grade/projective-dimension equality with simplicity of the Ext module characterizes Auslander-Gorenstein among 2-Gorenstein Nakayama algebras.

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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

First thing you should know: this is a real result, not a publicity job. The (G_n) criterion (Theorem 3.3) is short, correct, and useful, and the Nakayama application gives a clean reduction formula plus the odd-Ext zero-or-simple theorem, which does settle the Klász–Kleinau–Marczinzik conjecture when paired with their Proposition 5.2. I checked the core arguments the reader listed, and they hold. The paper is also honest about what is known: it credits the commutative case to Reiten–Fossum, presents Corollary 3.8 as a new proof of [14, Theorem 3.3], and flags the two-sentence generality transfer in Remark 4.1.

Two things to discharge before I would treat the full generality as gospel. First, Remark 4.1: the syzygy-filtration machinery from [22] was stated for algebras over an algebraically closed field; the paper says the proofs only use uniseriality and mutually isomorphic endomorphism rings of simples. That is likely true, but it is load-bearing for every theorem in Sections 4 and 5, and it is asserted rather than proved. A referee should ask for a proof or a restriction to split/algebraically closed settings. Second, Theorem 5.1(2) contains the line "T is a simple object of E_r" with no proof and no specific citation. The stress-test note is right that this is the most immediate internal gap. It is probably true by induction from the base-module construction, but the paper needs to spell it out or point to the exact statement in [22]. Also, the headline settlement depends on [13, Proposition 5.2], which is unpublished and not reproduced; transparent, but the reader should weigh a two-paper dependency.

None of these are demonstrated flaws. The mathematics I fully checked is correct, the literature is cited squarely, and the paper does not overclaim. It is worth a serious referee: I would send it out and ask the referee to focus on Section 5 and Remark 4.1 rather than on the criterion itself. I would cite this if I worked on Nakayama algebras; for a reading group I would probably wait until the loose ends are patched, but it is close.

Referee Report

2 major / 3 minor

Summary. The paper introduces a Serre-type condition (G_n) for two-sided Noetherian rings, expressed in terms of occurrence degrees and flat dimensions of indecomposable injectives, and proves in Theorem 3.3 that it is equivalent to the ring being n-Gorenstein. For Artin algebras this becomes a criterion relating grades of simple modules to projective dimensions of their injective envelopes. The paper then applies this criterion to Nakayama algebras using syzygy filtration, proving a reduction theorem (Theorem 4.12): A is n-Gorenstein iff A is 2-Gorenstein and ε(A) is (n−2)-Gorenstein. From this, the author derives preservation and reflection of the Auslander–Gorenstein property under syzygy filtration. The main application is Theorem 5.1, which asserts that over an Auslander–Gorenstein Nakayama algebra every odd Ext module Ext^n_A(S,A) is either zero or a simple A^op-module, and is nonzero exactly when pdim_A S = n. Together with a result of Klász–Kleinau–Marczinzik (Lemma 5.3), this gives Corollary 5.4 that simple modules of odd grade are regular, settling a recent conjecture.

Significance. If fully correct, the paper provides a clean and broadly applicable characterization of n-Gorenstein rings and a powerful reduction mechanism for Nakayama algebras. Theorem 3.3 is elegant and its proof is self-contained; the reduction Theorem 4.12 and the selfinjective-dimension formula in Theorem 4.14 are substantial original results. The final theorem on odd Ext modules is striking and would indeed settle the Klász–Kleinau–Marczinzik conjecture, as well as give a multiplicity-free statement for odd terms of the minimal injective coresolution. The exposition is clear, and the core homological steps in Sections 2–3 and the induction in Theorem 5.1(1) are carefully argued. However, two load-bearing points—the extension of the syzygy-filtration machinery to Artin algebras, and the simplicity assertion in the proof of Theorem 5.1(2)—are not adequately supported in the manuscript.

major comments (2)

[§5, proof of Theorem 5.1(2)] In the final paragraph of the proof, after writing n=2r+1 and T=Ω^{2r}(S), the paper states: 'By the construction of iterated syzygy filtration, T is a simple object of E_r.' No proof or precise citation is given. This assertion is not immediate: it requires an inductive proof that the second syzygy of a simple object of E_i, when its projective dimension is not 1, is a base module (hence a simple object) of E_{i+1}. The subsequent iteration of Lemma 5.2 depends entirely on this. Since part (2) is the basis for Corollaries 5.4 and 5.5, the proposed settlement of the conjecture is unsupported unless this statement is proved.
[Remark 4.1 and §4] The paper extends the syzygy-filtration results of [22] from finite-dimensional split algebras over an algebraically closed field to all basic Artin Nakayama algebras satisfying (N1)–(N2), with only the two-sentence justification in Remark 4.1. This is load-bearing: Lemma 4.2 (E is exact abelian, equivalent to mod ε(A), with Ext^E_* ≅ Ext^A_* and pdim_E = pdim_A), Proposition 4.4, Corollary 4.10, and Theorems 4.12–5.1 all rely on those results. The author should either prove the needed statements for Artin algebras or supply a precise theorem from [22] stated at that level of generality. As written, the validity of Sections 4–5 depends on an unverified assumption.

minor comments (3)

[Corollary 5.5] The proof asserts that for a Nakayama algebra the homothety homomorphism f: Δ_S → End_{A^op}(E) is an isomorphism. This is not obvious from simplicity of E alone, since a simple module can have dimension greater than one over its endomorphism ring. A justification or reference is needed for the multiplicity formula μ_n(S)=1.
[Definition 3.1 / Remark 3.1] It would help to state explicitly how (G_n) relates to the inequality established in Theorem 2.4: for module-finite algebras, the condition is only nontrivial in the direction not already guaranteed, which would clarify the formulation.
[Miscellaneous notation] Small typographical and notation issues: in the proof of Theorem 5.1(2), 'DTrT' should be 'DTr T'; in Corollary 4.13 the notation ε^i(A) might be typeset consistently. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's core reductions are proved from definitions and prior published machinery; the two fragile assertions are correctness gaps, not by-construction circularity.

full rationale

The claimed derivations do not reduce to their inputs by construction. Theorem 3.3 is proved directly from the definitions of n-Gorenstein and (G_n): if an indecomposable injective I first occurs at degree g<n, then I is a summand of I^g(R), so fdim I <= fdim I^g(R) <= g = occ_R(I); the converse reverses the same inequality. Theorem 3.5 is then a formal consequence via Lemma 2.2. Section 4 is the only place where the author's own prior work [22] is load-bearing: Remark 4.1 explicitly asserts that results stated for finite-dimensional algebras over an algebraically closed field transfer to basic Artin Nakayama algebras satisfying (N1)-(N2) because the proofs use only uniseriality and mutually isomorphic endomorphism rings of simples. This is a transfer claim about an independent published construction, not an equation identifying a predicted quantity with a fitted input; whether the transfer is correct is a correctness risk, not a circularity. In Theorem 5.1(2), the sentence 'By the construction of iterated syzygy filtration, T is a simple object of E_r' is asserted rather than proved and is load-bearing for the simplicity of odd Ext modules; but it is a lemma-level gap, not a reduction of Theorem 5.1 to its own conclusion. The simplicity of T in E_r is a statement about the syzygy-filtration construction, and the subsequent induction uses Lemma 5.2 and Theorem 4.14, both of which have independent proofs. No fitted parameters, no definition-in-terms-of-conclusion, and no uniqueness theorem imported from the authors are present. Hence no circular step can be exhibited under the required standard.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

Pure mathematics: no numbers are fitted, so the free-parameter list is empty — the only parameters in play are the integer n and the homological invariants defined by the theory. No new objects are postulated: the syzygy filtered algebra ε(A), the base modules ∇(S), and the filtration category E are constructions from [19–22], so the invented-entities list is empty. The genuine input of the paper is small: Theorem 3.3 costs only Lemma 2.2 plus the Faith–Walker decomposition, while the Nakayama half (Theorems 4.12–5.1) is a reduction to published machinery with one asserted generalization (Remark 4.1) and one imported preprint lemma ([13, Prop 5.2]).

assumptions (10)

standard math Every injective module over a left Noetherian ring is a direct sum of indecomposable injective modules (Faith–Walker).
Invoked in the '⇐=' direction of Theorem 3.3 to reduce the flat dimension of I^g(R) to its indecomposable summands; cited as [6].
standard math Govorov–Lazard: flat modules are filtered direct limits of finitely generated free modules, and Ext^*(X,−) commutes with filtered direct limits for finitely generated X over a left Noetherian ring.
Needed for Lemma 2.1, the engine of the inequality grade S ≤ fdim I(S); cited as [17].
domain assumption Lemma 2.3 (Skryabin/Bass): for a module-finite k-algebra R and prime P with p = P∩k, occ_R I_P = occ_{R_p}(I_P)_p and localization preserves injective envelopes.
Basis of Theorem 2.4 (occ ≤ fdim for module-finite algebras); cited from [23, Lemma 5.1] and [3, Cor. 1.3], not proved in the paper.
standard math Reiten–Fossum local characterization of commutative n-Gorenstein rings (Lemma 3.2).
Recalled to motivate (G_n); the paper proves (2)⟺(3) itself and cites [18] for (1)⟺(3).
standard math Ringel's Nakayama facts (Lemma 4.1): pdim S ≠ 1 iff P(τS) injective; idim S ≠ 1 iff I(τ^{-1}S) projective; im γ = {simples with idim ≠ 1}; γ bijects pdim ≠ 1 to idim ≠ 1.
Foundation of the base-module construction and of Lemmas 4.3/5.2; cited as [20, A.6].
domain assumption Syzygy filtration machinery of [22]: base modules form a semibrick; E is an exact abelian subcategory equivalent to mod ε(A); E contains second syzygies; Ext^E_* ≅ Ext^A_* and pdim_E = pdim_A on E.
The engine of Sections 4–5; assumed to hold for basic Artin Nakayama algebras with (N1)–(N2) by Remark 4.1, which asserts (not proves) that the algebraically-closed-field proofs of [22] carry over.
standard math 2-Gorenstein ⟺ 3-Gorenstein for Nakayama algebras (Fuller–Iwanaga, [8, 2.10]).
Used in Lemma 4.5 and Corollary 4.13 to handle the odd final step of the reduction; cited, not proved.
domain assumption [13, Proposition 5.2]: over an Auslander–Gorenstein Nakayama algebra, simple modules of odd grade are perfect (grade = pdim).
Imported from an unpublished preprint (Klász–Kleinau–Marczinzik, arXiv:2604.02146) in Lemma 5.3; without it Corollary 5.4 would only prove that odd-grade simples have simple Ext^n, not the regularity (grade = pdim plus Ext simple) stated in the conjecture.
standard math Zaks [25]: a two-sided Noetherian ring with finite left and right selfinjective dimension has equal left and right selfinjective dimensions.
Used in Theorem 4.14 to convert Corollary 4.7 (right dimension) into the two-sided formula vdim ε(A) = max{vdim A − 2, 0}.
standard math Left–right symmetry of the n-Gorenstein property (Fossum–Griffith–Reiten, [7, Thm 3.7]).
Lemma 3.1; used in Lemma 4.5(4) and Corollary 4.7.

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Pith. "Pith review of A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras." pith.science (2026). https://pith.science/paper/N2R4WCDN

@misc{pith2026260717497,
  author       = {Pith},
  title        = {Pith review of: A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2R4WCDN}},
  note         = {Machine review of arXiv:2607.17497}
}

abstract

Let $R$ be a left and right Noetherian ring. We introduce a Serre-type condition $(G_n)$, formulated in terms of the first occurrence and the flat dimension of indecomposable injective modules, and prove that $R$ is $n$-Gorenstein if and only if it satisfies $(G_n)$. We then apply the criterion to Nakayama algebras via syzygy filtration. It is shown that syzygy filtration preserves the Auslander-Gorenstein property and reflects it within the class of $2$-Gorenstein Nakayama algebras. Combined with a result of Kl\'asz, Kleinau and Marczinzik, this shows that simple modules of odd grade over Auslander-Gorenstein Nakayama algebras are regular in their grade, thereby settling their conjecture.

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Works this paper leans on

25 extracted references · 3 linked inside Pith

[22]

Shen,The resolution quiver of Nakayama algebras which are minimal Auslander- Gorenstein, J

D. Shen,The resolution quiver of Nakayama algebras which are minimal Auslander- Gorenstein, J. Algebra 701 (2026), 91-109

2026
[1]

Auslander, I

M. Auslander, I. Reiten,k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994), 1-27

1994
[2]

Auslander, I

M. Auslander, I. Reiten, S. O. Smalø,Representation theory of Artin algebras, Cambridge Stud. Adv. Math., 36. Cambridge University Press, Cambridge, 1995

1995
[3]

Bass,Injective dimension in noetherian rings, Trans

H. Bass,Injective dimension in noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18-29

1962
[4]

Bass,On the ubiquity of Gorenstein rings, Math

H. Bass,On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28

1963
[5]

Bruns, J

W. Bruns, J. Herzog,Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, Vol. 39. Cambridge University Press, Cambridge, 1998

1998
[6]

Faith and E

C. Faith and E. A. Walker,Direct-sum representations of injective modules, J. Algebra 5 (1967), 203-221

1967
[7]

R. M. Fossum, P. A. Griffith, I. Reiten,Trivial extensions of abelian categories, Homological algebra of trivial extensions of abelian categories with applications to ring theory. Lecture Notes in Math., Vol. 456. Springer-Verlag, Berlin-New York, 1975

1975

Show all 25 references

[8]

K. R. Fuller, Y. Iwanaga,Onn-Gorenstein rings and Auslander rings of low injective di- mension, In Representations of algebras (Ottawa, ON, 1992), 175-183, volume 14 of CMS Conf. Proc. Amer. Math. Soc., Providence, RI, 1993

1992
[9]

Gabriel,Des cat´ egories ab´ eliennes, Bull

P. Gabriel,Des cat´ egories ab´ eliennes, Bull. Soc. Math. France 90 (1962), 323-448

1962
[10]

Iyama,Auslander correspondence, Adv

O. Iyama,Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51-82

2007
[11]

Iyama, R

O. Iyama, R. Marczinzik,Distributive lattices and Auslander regular algebras, Adv. Math. 398 (2022), Paper No. 108233

2022
[12]

Kl´ asz,The Auslander-Gorenstein condition for monomial algebras, arXiv:2508.06957

V. Kl´ asz,The Auslander-Gorenstein condition for monomial algebras, arXiv:2508.06957

arXiv
[13]

Kl´ asz, M

V. Kl´ asz, M. Kleinau, R. Marczinzik,Classification of Auslander-Gorenstein monomial al- gebras: The acyclic case, arXiv:2604.02146
[14]

Kl´ asz, R

V. Kl´ asz, R. Marczinzik, H. Thomas,Auslander regular algebras and Coxeter matrices, arXiv:2501.09447

arXiv
[15]

H. Ko, V. Mazorchuk, R. Mrden,Some homological properties of categoryO. VI, Doc. Math. 26 (2021), 1237-1269

2021
[16]

T. Y. Lam,Lectures on modules and rings, Grad. Texts in Math., 189. Springer-Verlag, New York, 1999

1999
[17]

Lazard,Autour de la platitude, Bull

D. Lazard,Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81-128

1969
[18]

Reiten, R

I. Reiten, R. Fossum,Commutativen-Gorenstein rings, Math. Scand. 31 (1972), 33-48

1972
[19]

C. M. Ringel,The Gorenstein projective modules for the Nakayama algebras. I, J. Algebra 385 (2013), 241-261

2013
[20]

C. M. Ringel,The finitistic dimension of a Nakayama algebra, J. Algebra 576 (2021), 95-145

2021
[21]

Sen,Syzygy filtrations of cyclic Nakayama algebras, arXiv:1903.04645

E. Sen,Syzygy filtrations of cyclic Nakayama algebras, arXiv:1903.04645

1903 arXiv
[23]

Skryabin,On Gorensteinness of Hopf module algebras, Glasgow Math

S. Skryabin,On Gorensteinness of Hopf module algebras, Glasgow Math. J. 59 (2017), 299- 321

2017
[24]

Van Oystaeyen,Algebraic geometry for associative algebras, Monographs and Textbooks in Pure and Applied Mathematics, Vol

F. Van Oystaeyen,Algebraic geometry for associative algebras, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 232. Marcel Dekker, Inc., New York, 2000

2000
[25]

Zaks,Injective dimension of semi-primary rings, J

A. Zaks,Injective dimension of semi-primary rings, J. Algebra 13 (1969), 73-86. School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, P. R. China Email address:sdw12345@mail.ustc.edu.cn

1969

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [22]

Shen,The resolution quiver of Nakayama algebras which are minimal Auslander- Gorenstein, J

D. Shen,The resolution quiver of Nakayama algebras which are minimal Auslander- Gorenstein, J. Algebra 701 (2026), 91-109

2026

[2] [1]

Auslander, I

M. Auslander, I. Reiten,k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994), 1-27

1994

[3] [2]

Auslander, I

M. Auslander, I. Reiten, S. O. Smalø,Representation theory of Artin algebras, Cambridge Stud. Adv. Math., 36. Cambridge University Press, Cambridge, 1995

1995

[4] [3]

Bass,Injective dimension in noetherian rings, Trans

H. Bass,Injective dimension in noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18-29

1962

[5] [4]

Bass,On the ubiquity of Gorenstein rings, Math

H. Bass,On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28

1963

[6] [5]

Bruns, J

W. Bruns, J. Herzog,Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, Vol. 39. Cambridge University Press, Cambridge, 1998

1998

[7] [6]

Faith and E

C. Faith and E. A. Walker,Direct-sum representations of injective modules, J. Algebra 5 (1967), 203-221

1967

[8] [7]

R. M. Fossum, P. A. Griffith, I. Reiten,Trivial extensions of abelian categories, Homological algebra of trivial extensions of abelian categories with applications to ring theory. Lecture Notes in Math., Vol. 456. Springer-Verlag, Berlin-New York, 1975

1975