[1] H. Ananthnarayan, The Gorenstein colength of an Artinian local ring. J. Algebra 320 (2008), no. 9, 3438-–3446.
[2] H. Ananthnarayan; L.L. Avramov; W.F. Moore, Connected sums of Gorenstein local rings. J. Reine Angew. Math. 667 (2012), 149—176.
[3] T. Araya; K.-i. Iima; R. Takahashi, On the structure of Cohen–Macaulay modules over hypersurfaces of countable Cohen–Macaulay representation type, J. Algebra 361 (2012), 213–224.
[4] S. Ariki; R. Kase; K. Miyamoto, On components of stable Auslander–Reiten quivers that contain Heller lattices: the case of truncated polynomial rings, Nagoya Math. J. 228 (2017), 72–113.
[5] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5, (1961) 631–647.
[6] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, 511–531.
[7] M. Auslander; M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94, American Mathematical Society, Providence, R.I., 1969.
[8] M. Auslander; R.-O. Buchweitz, The homological theory of maximal Cohen–Macaulay approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987), M´em. Soc. Math. France (N.S.) 38 (1989), 5–37.
[9] M. Auslander; S. Ding; Ø. Solberg, Liftings and weak liftings of modules, J. Algebra 156 (1993), 273–317.
[10] L. L. Avramov, Infinite free resolutions, Six lectures on commutative algebra, 1–118, Mod. Birkh¨auser Class., Birkh¨auser Verlag, Basel, 2010.
[11] L. L. Avramov, A cohomological study of local rings of embedding codepth 3, J. Pure Appl. Algebra 216 (2012), no. 11, 2489–2506.
[12] L. L. Avramov; S. B. Iyengar, Constructing modules with prescribed cohomological support, Illinois J. Math. 51 (2007), no. 1, 1–20.
[13] L. L. Avramov; S. Iyengar; L. S¸ega, Free resolutions over short local rings, Proc. London Math. Soc. 78, (2008) 459–476.
[14] H. Bass, On the ubiquity of Gorenstein rings, Math. Zeitschrift 82 (1963), 8–28.
[15] V. Barucci and R. Froberg ¨ , One-dimensional almost Gorenstein rings, J. Algebra, 188 (1997), no. 2, 418–442.
[16] G. M. Bergman, Minimal faithful modules over Artinian rings. Publ. Mat. 59 (2015), no. 2, 271-–300.
[17] J. P. Brennan; J. Herzog; B. Ulrich, Maximally generated Cohen–Macaulay modules, Math. Scand. 61 (1987), no. 2, 181–203.
[18] J. Brennan; W. Vasconcelos, On the structure of closed ideals, Math. Scand. 88 (2001), no. 1, 3–16.
[19] W. Bruns; J. Herzog, Cohen–Macaulay rings, revised edition, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998.
[20] R.-O. Buchweitz, Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, unpublished paper (1986), http://hdl.handle.net/1807/16682.
[21] R.-O. Buchweitz; G.-M. Greuel; F.-O. Schreyer, Cohen–Macaulay modules on hypersurface singularities, II, Invent. Math. 88 (1987), no. 1, 165–182.
[22] I. Burban; Y. Drozd, Maximal Cohen–Macaulay modules over non-isolated surface singularities and matrix problems, Mem. Amer. Math. Soc. 248 (1178) (2017).
[23] L. Burch, On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc., 64 (1968), 941–948.
[24] L. Burch, A note on the homology of ideals generated by three elements in local rings, Proc. Cambridge Philos. Soc., 64 (1968), 949–952.
[25] L. Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc., 72 (1972), 369–373.
[26] M. Casanellas; R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 709–731.
[27] O. Celikbas, Vanishing of Tor over complete intersections, J. Commut. Algebra 3 (2011),169–206.
[28] O. Celikbas; S. Goto; R. Takahashi; N. Taniguchi, On the ideal case of a conjecture of Huneke and Wiegand, to appear in Proc. Edinb. Math. Soc. (2).
[29] O. Celikbas; K. Iima; A. Sadeghi; R. Takahashi, On the ideal case of a conjecture of Auslander and Reiten,Bull. Sci. Math. 142 (2018), 94–107.
[30] O. Celikbas; R. Takahashi, Auslander–Reiten conjecture and Auslander–Reiten duality, J. Algebra 382 (2013), 100–114.
[31] S. Choi, Exponential growth of Betti numbers, J.Algebra 152 (1992), 20–29.
[32] L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.
[33] A. Conca; E. DeNegri; M. E. Rossi, Integrally closed and componentwise linear ideals, Math. Z. 265 (2010), 715–734.
[34] A. Corso; S. Goto; C. Huneke; C. Polini; B. Ulrich, Iterated socles and integral dependence in regular rings, Trans. Amer. Math. Soc. 370 (2018), no. 1, 53–72.
[35] A. Corso; C. Huneke; W. V. Vasconcelos, On the integral closure of ideals, Manuscripta Math. 95 (1998), no. 3, 331–347.
[36] A. Corso; C. Huneke; D. Katz; W. V. Vasconcelos, Integral closure of ideals and annihilators of homology, Lect. Notes Pure Appl. Math. 204 (2006), 33–48.
[37] L. Costa; R. M. Miro-Roig ´ , GL(V )-invariant Ulrich bundles on Grassmannians, Math. Ann. 361 (2015), no. 1-2, 443–457.
[38] D. T. Cuong, Problem Session, International Workshop on Commutative Algebra, Thai Nguyen University, January 4, 2017.
[39] C. W. Curtis;I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. 1, Wiley-Interscience, New York, 1981.
[40] H. Dao; O. Iyama; R. Takahashi; C. Vial, Non-commutative resolutions and Grothendieck groups, J. Noncommut. Geom. 9 (2015), no. 1, 21–34.
[41] H. Dao; T. Kobayashi; R. Takahashi, Burch ideals and Burch rings, preprint, arXiv:1905.02310.
[42] H. Dao; J. Schweig, The type defect of a simplicial complex, Journal of Combinatorial Theory, Series A 163 (2019), 195–210.
[43] H. Dao; R. Takahashi, The dimension of a subcategory of modules, Forum Math. Sigma 3 (2015), e19, 31 pp.
[44] A. De Stefani, Products of ideals may not be Golod, J. Pure Appl. Algebra, 220 (2016), no. 6, 2289–2306.
[45] M. T. Dibaei; M. Rahimi, Rings with canonical reduction. arXiv:1712.00755.
[46] Y. Drozd, Representations of commutative algebras, Funct. Analysis and its Appl., 6 (1972), English Translation, 286–288.
[47] Y. Drozd; V. V. Kiricenko ˇ The quasi-Bass orders, (Russian) Izv. Akad.Nauk SSSR Ser. Mat. 36 (1972), 328–370.
[48] Y. Drozd; A. Ro˘ıter, Commutative rings with a finite number of indecomposable integral representations, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 783–798.
[49] S. P. Dutta, Syzygies and homological conjectures, Commutative algebra (Berkeley, CA, 1987), 139–156, Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989.
[50] D. Eisenbud, Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
[51] E. G. Evans; P. Griffith, Syzygies, London Mathematical Society Lecture Note Series, 106,
[52] J. Elias; M. Silva Takatuji, On Teter rings. Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), no. 1, 125—139.
[53] H. Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63–71.
[54] R. M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973.
[55] P. Gabriel, Des cat´egories ab´eliennes, Bull. Soc. Math. France 90 (1962), 323–448.
[56] P. Gabriel, Unzerlegbare Darstellungen, I, Manuscripta math. 6 (1972), 71–103.
[57] L. Ghezzi; S.Goto; J. Hong; W. V. Vasconcelos, Invariants of Cohen–Macaulay rings associated to their canonical ideals. Journal of Algebra, 489, 506–528.
[58] S. Goto; R. Isobe; S. Kumashiro, Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings, to appear in J. Pure Appl. Algebra.
[59] P. A. Garc´ıa-Sanchez; M. J. Leamer ´ , Huneke-Wiegand conjecture for complete intersection numerical semigroup rings, J. Algebra 391 (2013), 114–124.
[60] S. Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151–160.
[61] S. Goto; F. Hayasaka, Finite homological dimension and primes associated to integrally closed ideals, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3159–3164.
[62] S. Goto; N. Matsuoka; T. T. Phuong, Almost Gorenstein rings, J. Algebra, 379 (2013), 355–381.
[63] S. Goto; K. Ozeki; R. Takahashi; K.-I. Watanabe; K.-I. Yoshida, Ulrich ideals and modules, Math. Proc. Cambridge Philos. Soc. 156 (2014), no. 1, 137–166.
[64] S. Goto; K. Ozeki; R. Takahashi; K.-I. Watanabe; K.-I. Yoshida, Ulrich ideals and modules over two-dimensional rational singularities, Nagoya Math. J. 221 (2016), no. 1, 69–110.
[65] S. Goto; R. Takahashi; N. Taniguchi, Almost Gorenstein rings – towards a theory of higher dimension, J. Pure Appl. Algebra 219 (2015), no. 7, 2666–2712.
[66] S. Goto; R. Takahashi; N. Taniguchi; H. Le Truong, Huneke-Wiegand conjecture and change of rings, J. Algebra 422 (2015), 33–52.
[67] E. Green; I. Reiner, Integral representations and diagrams, Michigan Math. J. 25 (1978), no. 1, 53–84.
[68] Tor H. Gulliksen, On the length of faithful modules over Artinian local rings, Math. Scand. 31 (1972) 78–82.
[69] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119. Cambridge University Press, Cambridge, 1988. x+208 pp.
[70] W. J. Heinzer; L. J. Ratliff Jr. ; D. E. Rush, Basically full ideals in local rings, J. Algebra 250 (2002), no. 1, 371–396.
[71] I. Henriques, L. S¸ega, Free resolutions over short Gorenstein local rings, Math. Z. 267 (2011), 645–663.
[72] K. Herzinger, The number of generators for an ideal and its dual in a numerical semigroup, Comm. Algebra 27 (1999), 4673–4690.
[73] J. Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen–Macaulay–Moduln, Math. Ann. 233 (1978), no. 1, 21–34.
[74] J. Herzog; T. Hibi; D. Stamate, The trace of the canonical module, Israel J. Math. 233 (2019), no. 1, 133––165.
[75] J.Herzog; A. Simis; W. Vasconcelos, Approximations complexes and blowing-up rings II, J. Algebra 82(l) (1983),53–83.
[76] J. Herzog; B. Ulrich; J. Backelin, Linear maximal Cohen–Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), no. 2-3, 187–202.
[77] H. Hijikata; K. Nishida, Bass orders in nonsemisimple algebras, J. Math. Kyoto Univ. 34 (1994), no. 4, 797–837.
[78] J. Hong; H. Lee; S. Noh; D. E. Rush, Full ideals, Comm. Algebra 37 (2009), no. 8, 2627–2639.
[79] M. J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985), 73–96, London Math. Soc. Lecture Note Ser., 117,Cambridge Univ. Press, Cambridge, 1987.
[80] C. Huneke; S. Iyengar; R. Wiegand, Rigid Ideals in Gorenstein Rings of Dimension One. Acta Mathematica Vietnamica (2018), 1–19.
[81] C. Huneke; D. A. Jorgensen, Symmetry in the vanishingof Ext over Gorenstein rings, Math. Scand. 93 (2003), 161–184.
[82] C. Huneke; G. J. Leuschke, Two theorems about maximal Cohen–Macaulay modules, Math. Ann. 324 (2002), no. 2, 391–404.
[83] C. Huneke; G. J. Leuschke, Local rings of countable Cohen–Macaulay type, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3003–3007.
[84] C. Huneke; G. J. Leuschke, On a conjecture of Auslander and Reiten, J. Algebra 275 (2004), no. 2, 781–790.
[85] C. Huneke; I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006.
[86] C. Huneke; A. Vraciu, Rings that are almost Gorenstein, Pacific J. Math. 225 (2006), no. 1, 85–102.
[87] C. Huneke; R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), no. 3, 449–476.
[88] O. Iyama; M. Wemyss, Maximal modifications and Auslander–Reiten duality for nonisolated singularities, Invent. Math. 197 (2014), no. 3, 521–586.
[89] S. B. Iyengar; R. Takahashi, Annihilation of cohomology and strong generation of module categories, Int. Math. Res. Not. IMRN 2016, no. 2, 499–535.
[90] H. Jacobinski, Sur les ordres commutatifs avec un nombre fini de r´eseaux ind´ecomposables, Acta Math. 118 (1967), 1–31.
[91] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379–417.
[92] J. O. Kleppe; R. M, Miro-Roig ´ , On the normal sheaf of determinantal varieties, J. Reine Angew. Math. 719 (2016), 173–209.
[93] L. Klingler; L. Levy, Representation type of commutative Noetherian rings. I. Local wildness, Pacific J. Math. 200 (2001), no. 2, 345–386.
[94] T. Kobayashi, Syzygies of Cohen–Macaulay modules and Grothendieck groups, J. Algebra 490 (2017), 372–379.
[95] T. Kobayashi, Syzygies of Cohen–Macaulay modules over one dimensional Cohen– Macaulay local rings, preprint, arXiv:1710.02673.
[96] T. Kobayashi, The Huneke-Wiegand conjecture and middle terms of almost split sequences, preprint, arXiv:1907.02348.
[97] T. Kobayashi, Local rings with a self-dual maximal ideal, preprint, arXiv:1812.10341.
[98] T. Kobayashi; J. Lyle; R. Takahashi, Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum, preprint, arXiv:1903.03287.
[99] T. Kobayashi; R. Takahashi, Ulrich modules over Cohen-Macaulay local rings with minimal multiplicity, Q. J. Math. 70 (2019), no. 2, 487–507.
[100] T. Kobayashi; R. Takahashi, Rings whose ideals are isomorphic to trace ideals, Math. Nachr. 292 (2019), no. 10, 2252–2261.
[101] H. Knorrer ¨ , Cohen–Macaulay modules on hypersurface singularities, I, Invent. Math. 88 (1987), no. 1, 153–164.
[102] A. I. Kostrikin; I. R.Safarevi ˘ c˘, Groups of homologies of nilpotent algebras, Dokl. Akad.Nauk SSSR 115 (1957), 1066–1069.
[103] H. Krause and G. Stevenson, A note on thick subcategories of stable derived categories, Nagoya Math. J. 212 (2013), 87–96.
[104] E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970), 748–751.
[105] A. R. Kustin; A. Vraciu, Totally reflexive modules over rings that are close to Gorenstein, J. Algebra (to appear), arXiv:1705.05714v1.
[106] J. Lescot, La s´erie de Bass d’un produit fibr´e d’anneaux locaux, C. R. Acad. Sci. Paris S´er. IMath. 293 (1981), no. 12, 569–571.
[107] G. J. Leuschke; R. Wiegand, Local rings of bounded Cohen–Macaulay type, Algebr. Represent. Theory 8 (2005), no. 2, 225–238.
[108] G. J. Leuschke; R. Wiegand, Cohen–Macaulay Representations, Mathematical Surveys and Monographs, vol. 181, American Mathematical Society, Providence, RI, 2012.
[109] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10, (1966) 220–226.
[110] H. Lindo, Trace ideals and centers of endomorphism rings of modules over commutative rings, J. Algebra 482 (2017), 102–130.
[111] H. Lindo; N. Pande, Trace ideals and the Gorenstein property, arXiv:1802.06491v2.
[112] J. Lipman, Stable ideals and Arf rings, Amer. J. Math., 93 (3), 649–685.
[113] H. Matsui; R. Takahashi; Y. Tsuchiya, When are n-syzygy modules n-torsionfree?, Arch. Math. (Basel) 108 (2017), no. 4, 351–355.
[114] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid, Second edition, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989.
[115] Y. Nakajima; K.-i. Yoshida, Ulrich modules over cyclic quotient surface singularities, J. Algebra 482 (2017), 224–247.
[116] S. Nasseh, R. Takahashi, Local rings with quasi-decomposable maximal ideal, Math. Proc. Cambridge Philos. Soc. (to appear), arXiv:1704.00719v2.
[117] A. Neeman, The chromatic tower for D(R), With appendix by Marcel B¨okstedt, Topology 31 (1992), no. 3, 519–532.
[118] J.-i. Nishimura, A few examples of local rings, I, Kyoto J. Math. 52 (2012), no. 1, 51–87.
[119] T. Ogoma, Cohen Macaulay factorial domain is not necessarily Gorenstein, Mem. Fac. Sci. Kˆochi Univ. Ser. A Math. 3 (1982), 65–74.
[120] T. Ogoma, Existence of dualizing complexes, J. Math. Kyoto Univ. 24 (1984), no. 1, 27–48.
[121] A. Ooishi, On the self-dual maximal Cohen–Macaulay modules, J. Pure Appl. Algebra 106 (1996), no. 1, 93–102.
[122] D. Quillen, Higher algebraic K-theory, I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147, Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973.
[123] K. Ringel, The representation type of local algebras, Springer Lecture Notes in Mathematics, 488 (1975), 282–305.
[124] J. C. Rosales, Numerical semigroups that differ from a symmetric numerical semigroup in one element. Algebra Colloq. 15 (2008), no. 1, 23-–32.
[125] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–256.
[126] R. Roy, Auslander–Reiten Sequences over Gorenstein Rings of Dimension One (2018). Dissertations - ALL. 873. https://surface.syr.edu/etd/873
[127] R. Roy, Graded AR Sequences and the Huneke–Wiegand Conjecture, arXiv:1808.06600.
[128] D. E. Rush, Contracted, m-full and related classes of ideals in local rings, Glasg. Math. J. 55 (2013), no. 3, 669–675.
[129] H. Sabzrou; F. Rahmati, The Frobenius number and a-invariant, Rocky Mountain J. Math. 36 (2006), no. 6, 2021–2026.
[130] J. D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 (1980), no. 2, 167–175.
[131] J. D. Sally, Cohen-Macaulay local rings of embedding dimension e+d - 2, J. Algebra 83 (1983), 393–408.
[132] S. Sather-Wagstaff, Semidualizing modules and the divisor class group, Illinois J. Math. 51 (2007), no. 1, 255–285.
[133] F.-O. Schreyer, Finite and countable CM-representation type, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 9–34, Lecture Notes in Math., 1273, Springer, Berlin, 1987.
[134] B. Stone, Super-stretched and graded countable Cohen–Macaulay type, J. Algebra 398 (2014), 1–20.
[135] J. Striuli; A. Vraciu, Some homological properties of almost Gorenstein rings, Commutative algebra and its connections to geometry, 201–215, Contemp. Math., 555, Amer. Math. Soc., Providence, RI, 2011.
[136] R. Takahashi, Syzygy modules with semidualizing or G-projective summands, J. Algebra 295 (2006), no. 1, 179–194.
[137] R. Takahashi, An uncountably infinite number of indecomposable totally reflexive modules, Nagoya Math. J. 187 (2007), 35–48.
[138] R. Takahashi, On G-regular local rings, Comm. Algebra 36 (2008), no. 12, 4472–4491.
[139] R. Takahashi,Modules in resolving subcategories which are free on the punctured spectrum. Pacific J. Math. 241 (2009), no. 2, 347–367.
[140] R. Takahashi, Classifying thick subcategories of the stable category of Cohen–Macaulay modules, Adv. Math. 225 (2010), no. 4, 2076–2116.
[141] R. Takahashi, Classifying resolving subcategories over a Cohen–Macaulay local ring, Math. Z. 273 (2013), no. 1-2, 569–587.
[142] R. Takahashi, Reconstruction from Koszul homology and applications to module and derived categories, Pacific J. Math. 268 (2014), no. 1, 231–248.
[143] W. Teter, Rings which are a factor of a Gorenstein ring by its socle. Inventione Math, 23 (1974), 153–162.
[144] B. Ulrich, Gorenstein rings and modules with high numbers of generators, Math. Z. 188 (1984), no. 1, 23–32.
[145] W. V. Vasconcelos, Ideals generated by R-sequences, J. Algebra 6 (1967), 309–316.
[146] J. Watanabe, m-full ideals, Nagoya Math. J. 106 (1987), 101–111.
[147] J. Watanabe, The syzygies of m-full ideals, Math. Proc. Cambridge Philos. Soc. 109 (1991), 7–13.
[148] K.-i. Yoshida, A note on minimal Cohen–Macaulay approximations, Comm. Algebra 24 (1996), no. 1, 235–246.
[149] Y. Yoshino, Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical SocietyLecture Note Series, 146,Cambridge University Press, Cambridge, 1990.