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Polynomial Identity Rings

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References

  1. S. Abeasis, M. Pittaluga, On a minimal set of generators for the invari-ants of 3 x 3 matrices, Comm Algebra 17 (1989), 487–499.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Alev, L. Le Bruyn, Automorphisms of generic 2 by 2 matrices, in “Perspectives in Ring Theory” (Antwerp, 1987), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 233 Kluwer Acad. Publ., Dordrecht, 1988, 6983.

    Google Scholar 

  3. S.A. Amitsur, The T-ideals of the free ring,J. London Math. Soc. 30(1955), 470–475.

    Article  MathSciNet  MATH  Google Scholar 

  4. S.A. Amitsur, A note on PI-rings, Israel J. Math. 10 (1971), 210–211.

    MathSciNet  MATH  Google Scholar 

  5. S.A. Amitsur, J. Levitzki, Minimal identities for algebras, Proc. Amer.Math. Soc. 1 (1950), 449–463.

    Article  MathSciNet  MATH  Google Scholar 

  6. S.A. Amitsur, A. RegevPI-algebras and their cocharacters, J. Algebra 78 (1982), 248–254.

    Article  MathSciNet  MATH  Google Scholar 

  7. S.A. Amitsur, L.W. Small, Affine algebras with polynomial identities,Rend. Circ. Mat. Palermo (2) Suppl. 31 (1993), 9–43.

    MathSciNet  Google Scholar 

  8. H. Aslaksen, Eng-Chye Tan, Chen-Bo Zhu, Generators and relations of invariants of 2 x 2 matrices, Comm. Algebra 22 (1994), 1821–1832.

    Article  MathSciNet  MATH  Google Scholar 

  9. A.Ya. Belov, On a Shirshov basis of relatively free algebras of complexity n (Russian), Mat. Sb. 135 (1988), 373–384. Translation: Math. USSR Sb. 63 (1988), 363–374.

    Article  Google Scholar 

  10. A.Ya. Belov About height theorem, Comm Algebra 23 (1995), 3551–3553.

    Article  MathSciNet  MATH  Google Scholar 

  11. A.Ya. Belov Rationality of Hilbert series of relatively free algebras (Russian), Uspekhi Mat. Nauk 52 (1997), No. 2, 153–154. Translation: Russian Math. Surveys 52 (1997), 394–395.

    Article  MathSciNet  Google Scholar 

  12. A.Ya. Belov, On non-Specht varieties (Russian), Fundam. Prikl. Mat. 5 (1999), No. 1, 47–66.

    MathSciNet  MATH  Google Scholar 

  13. A.Ya. Belov, Counterexamples to the Specht problem (Russian), Mat. Sb. 191 (2000), No. 3, 13–24. Translation: Sb. Math. 191 (2000), No. 3, 329–340.

    Article  MathSciNet  MATH  Google Scholar 

  14. A.Ya. Belov, V.V. Borisenko, V.N. Latyshev, Monomial algebras, Algebra, 4, J. Math. Sci. (New York) 87 (1997), No. 3, 3463–3575.

    Article  MathSciNet  MATH  Google Scholar 

  15. F. Benanti, J. Demmel, V. Drensky, P. Koev, Computational approach to polynomial identities of matrices - a survey, in “Ring Theory: PolynomialIdentities and Combinatorial Methods, Proc. of the Conf. in Pantelleria”; Eds. A. Giambruno, A. Regev, and M. Zaicev, Lect. Notes in Pure and Appl. Math. 235 Dekker, 2003, 141–178.

    Google Scholar 

  16. F. Benanti, V. Drensky, On the consequences of the standard polynomial,Comm Algebra 26 (1998), 4243–4275.

    Article  MathSciNet  MATH  Google Scholar 

  17. I.I. Benediktovich, A.E. Zalesskij, Almost standard identities of matrix algebras (Russian), Dokl. Akad. Nauk BSSR 23 (1979), 201–204.

    MathSciNet  MATH  Google Scholar 

  18. A. Berele, Homogeneous polynomial identities, Israel J. Math. 42 (1982),258–272.

    MathSciNet  MATH  Google Scholar 

  19. A. Berele, Magnum P.I, Israel J. Math. 51 (1985), 13–19.

    MATH  Google Scholar 

  20. A. Berele, Generic verbally prime PI-algebras and their GK-dimensions, Comm. Algebra 21 (1993), 1487–1504.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Berele, Rates and growth of PI-algebras, Proc. Amer. Math. Soc. 120 (1994), 1047–1048.

    MathSciNet  MATH  Google Scholar 

  22. A. Berele, Supertraces and matrices over Grassmann algebras, Adv.Math. 108 (1994), 77–90.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Berele, A. Regev, Applications of hook Young diagrams to P.I. alge-bras, J. Algebra 82 (1983), 559–567.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Berele, A. Regev, Codimensions of products and of intersections of verbally prime T-ideals, Israel J. Math. 103 (1998), 17–28.

    MathSciNet  MATH  Google Scholar 

  25. G.M. Bergman, A note on growth functions of algebras and semi-groups, Research Note, Univ. of California, Berkeley, 1978, unpublished mimeographed notes.

    Google Scholar 

  26. G.M. Bergman, Wild automorphisms of free P.I. algebras, and some new identities, preprint.

    Google Scholar 

  27. S. BondariConstructing the polynomial identities and central identities of degree < 9 of 3 x 3 matrices, Lin. Algebra Appl. 258 (1997), 233–249.

    Article  MathSciNet  MATH  Google Scholar 

  28. W. Borho, H. KraftÜber die Gelfand-Kirillov Dimension, Math. Ann 20 (1976), 1–24.

    Article  MathSciNet  Google Scholar 

  29. A. BraunThe nilpotency of the radical in a finitely generated PI-ring,J. Algebra 89 (1984), 375–396.

    Article  MathSciNet  MATH  Google Scholar 

  30. W. BurnsideOn an unsettled question in the theory of discontinuous roups, Quart. J. Math. 33 (1902), 230–238.

    MATH  Google Scholar 

  31. L. CariniThe Poincaré series related to the Grassmann algebra, Lin.Multilin Algebra 27 (1990), 199–205.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Carini, O. M. Di Vincenzo, On the multiplicity of the cocharacters of the tensor square of the Grossmann algebra, Atti dell’Accademia Peloritana dei Pericolanti, Messina, Classe I di Scienze Fis. Mat. e Nat. 69 (1991), 237–246.

    MATH  Google Scholar 

  33. Q. ChangSome consequences of the standard identity, Proc. Amer.Math. Soc. 104 (1988), 707–710.

    Article  MathSciNet  MATH  Google Scholar 

  34. G.P.On local finiteness of algebras (Russian), Mat. Issled. 105(1988), 153–171.

    MathSciNet  MATH  Google Scholar 

  35. P.M. CohnFree Rings and Their Relations, Second Edition London ath. Soc. Monographs 19 Acad. Press, 1985.

    Google Scholar 

  36. C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and ssociative Algebras, Interscience, John Willey and Sons, New York-London, 1962.

    Google Scholar 

  37. C. De Concini, C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976), 330–354.

    Article  MathSciNet  Google Scholar 

  38. J.A. Dieudonnü, J.B. Carrell, Invariant Theory, Old and New, Academic ress, New York - London, 1971.

    Google Scholar 

  39. R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann.of Math. 51 (1950), 161–166.

    Article  MathSciNet  MATH  Google Scholar 

  40. O.M. Di Vincenzo, On the Kronecker product of S n - cocharacters for P. I. algebras, Lin. Multilin. Algebra 23 (1988), 139–143.

    MATH  Google Scholar 

  41. O.M. Di Vincenzo, A note on the identities of the Grossmann algebras, Boll. Un. Mat. Ital. (7) 5-A (1991), 307–315.

    MATH  Google Scholar 

  42. O.M. Di Vincenzo, On the graded identities of M 1 , l (E), Israel J. Math. 80 (1992), 323–335.

    Article  MathSciNet  MATH  Google Scholar 

  43. O.M. Di Vincenzo, V. Nardozza, \({{\Bbb Z}_{k + l}} \times {{\Bbb Z}_2}\)-graded polynomial identities for \({M_k}{,_l}(E) \otimes E\), Rend. Semin Mat. Univ. Padova 108 (2002), 27–39.

    MATH  Google Scholar 

  44. M. Domokos, Eulerian polynomial identities and algebras satisfying a standard identity,J. Algebra 169 (1994), 913–928.

    Article  MathSciNet  MATH  Google Scholar 

  45. M. DomokosNew identities for 3 x 3 matrices, Lin. Multilin. Algebra 38 (1995), 207–213.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Domokos, S.G. Kuzmin, A.N. Zubkov, Rings of matrix invariants in positive characteristic, J. Pure Appl. Algebra 176 (2002), 61–80.

    Article  MathSciNet  MATH  Google Scholar 

  47. P. Doubilet, G.C. Rota, J. Stein, On the foundations of combinatorial theory, IX. Combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216.

    MathSciNet  MATH  Google Scholar 

  48. V. Drensky, Representations of the symmetric group and varieties of linear algebras (Russian), Mat. Sb. 115 (1981), 98–115. Translation: Math. USSR Sb. 43 (1981), 85–101.

    MathSciNet  Google Scholar 

  49. V. Drensky, A minimal basis for the identities of a second-order matrix algebra over a field of characteristic 0 (Russian), Algebra i Logika 20 (1981), 282–290. Translation: Algebra and Logic 20 (1981), 188–194.

    Article  MathSciNet  Google Scholar 

  50. V. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras, J. Algebra 91 (1984), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Drensky, On the Hilbert series of relatively free algebras, Comm Al-gebra 12 (1984), 2335–2347.

    Article  MathSciNet  MATH  Google Scholar 

  52. V. Drensky, Extremal varieties of algebras. I, II (Russian), Serdica 13 (1987), 320–332; 14 (1988), 20–27.

    MathSciNet  MATH  Google Scholar 

  53. V. Drensky, Explicit formulas for the codimensions of some T-ideals (Russian), Sibirsk. Mat. Zh. 29 (1988) No. 6, 30–36. Translation: Siberian Math. J. 29 (1988), 897–902.

    MathSciNet  Google Scholar 

  54. V. Drensky, New central polynomials for the matrix algebra, Israel J. Math. 92 (1995), 235–248.

    MathSciNet  MATH  Google Scholar 

  55. V. Drensky, Commutative and noncommutative invariant theory, Math. and Education in Math. Proc. of 24th Spring Conf. of the Union of Bulg. Mathematicians, Sofia, 1995, 14–50.

    Google Scholar 

  56. V. Drensky, Gelfand-Kirillov dimension of PI-algebras, in “Methods in Ring Theory, Proc. of the Trento Conf.”, Lect. Notes in Pure and Appl. Math. 198 Dekker, 1998, 97–113.

    Google Scholar 

  57. V. Drensky, Free Algebras and PI-Algebras, Springer-Verlag, Singapore, 2000.

    MATH  Google Scholar 

  58. V. Drensky, Defining relations for the algebra of invariants of 2 x 2 ma- trices, Algebras and Representation Theory 6 (2003), 193–214.

    Article  MathSciNet  MATH  Google Scholar 

  59. V. Drensky, A. Kasparian, Polynomial identities of eighth degree for 3 x 3 matrices, Annuaire de l’Univ. de Sofia, Fac. de Math. et Mecan., Livre 1, Math. 77 (1983), 175–195.

    MathSciNet  Google Scholar 

  60. V. Drensky, A. Kasparian, A new central polynomial for 3 x 3 matrices, Comm Algebra 13 (1985), 745–752.

    Article  MathSciNet  Google Scholar 

  61. V. Drensky, P. Koshlukov, Weak polynomial identities for a vector space with a symmetric bilinear form, Math. and Education in Math. Proc. of 16-th Spring Conf. of the Union of Bulg. Mathematicians, Sofia, Publ. House of the Bulg. Acad. of Sci., 1987, 213–219.

    Google Scholar 

  62. V. Drensky, G.M. Piacentini Cattaneo, A central polynomial of low degree for 4 x 4 matrices, J. Algebra 168 (1994), 469–478.

    Article  MathSciNet  MATH  Google Scholar 

  63. V. Drensky, Ts.G. Rashkova, Weak polynomial identities for the matrix algebras, Comm. Algebra 21 (1993), 3779–3795.

    Article  MathSciNet  MATH  Google Scholar 

  64. J. Dubnov, Sur une généralisation de l’équation de Hamilton-Cayley et sur les invariants simultanés de plusieurs affineurs, Proc. Seminar on Vector and Tensor Analysis, Mechanics Research Inst., Moscow State Univ. 2/3 (1935), 351–367 (see also Zbl. für Math. 12 (1935), p. 176).

    Google Scholar 

  65. J. Dubnov, V. Ivanov, Sur l’abaissement du degrü des polynômes en affineurs, C.R. (Doklady) Acad. Sci. USSR 41 (1943), 96–98 (see also MR 6 (1945), p. 113, Zbl. für Math. 60 (1957), p. 33).

    MathSciNet  Google Scholar 

  66. E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129–132.

    Article  MathSciNet  MATH  Google Scholar 

  67. E. Formanek, The polynomial identities of matrices, Contemp. Math. 13 (1982), 41–79.

    MathSciNet  MATH  Google Scholar 

  68. E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), 178–223.

    Article  MathSciNet  MATH  Google Scholar 

  69. E. Formanek, Noncommutative invariant theory, Contemp. Math. 43 (1985), 87–119.

    MathSciNet  Google Scholar 

  70. E. Formanek, Functional equations for character series associated with n x n matrices, Trans. Amer. Math. Soc. 294 (1986), 647–663.

    MathSciNet  MATH  Google Scholar 

  71. E. Formanek, The Polynomial Identities and Invariants of n x n Matri-ces, CBMS Regional Conf. Series in Math. 78 Published for the Confer. Board of the Math. Sci. Washington DC, AMS, Providence RI, 1991.

    Google Scholar 

  72. E. Formanek, The ring of generic matrices, J. Algebra 258 (2002), 310–320.

    Article  MathSciNet  MATH  Google Scholar 

  73. G. Freundeburg, A survey of counterexamples to Hilbert’s fourteenth problem, Serdica Math. J. 27 (2001), 171–192.

    Google Scholar 

  74. A. Giambruno, S.K. Sehgal, On a polynomial identity for n x n matrices,J. Algebra 126 (1989), 451–453.

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Giambruno, A. Valenti, Central polynomials and matrix invariants,Israel J. Math. 96 (1996), 281–297.

    MathSciNet  MATH  Google Scholar 

  76. A. Giambruno, M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145–155.

    Article  MathSciNet  MATH  Google Scholar 

  77. A. Giambruno, M. Zaicev, Exponential codimension growth of PI alge- bras: An exact estimate, Adv. Math. 142 (1999), 221–243.

    Article  MathSciNet  MATH  Google Scholar 

  78. A. Giambruno, M. Zaicev, Minimal varieties of algebras of exponential growth, Adv. Math. 174 (2003), 310–323.

    Article  MathSciNet  MATH  Google Scholar 

  79. A. Giambruno, M. Zaicev, Codimension growth and minimal snperalge-bras,Trans. Amer. Math. Soc. 355 (2003), 5091–5117.

    Article  MathSciNet  MATH  Google Scholar 

  80. E.S. Golod, On nil-algebras and finitely approximable p-groups (Russian),Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 273–276.

    MathSciNet  Google Scholar 

  81. E.S. Golod, I.R. Shafarevich, On the class field tower (Russian), Izv.Akad. Nauk SSSR, Ser. Mat. 28 (1964), 261–272.

    MathSciNet  MATH  Google Scholar 

  82. A.V. Grishin, Examples of T-spaces and T-ideals over a field of charac-teristic 2 without the finite basis property (Russian, English summary), Fundam. Prikl. Mat. 5 (1999), No. 1, 101–118.

    MathSciNet  MATH  Google Scholar 

  83. A.V. Grishin, On non-Spechtianness of the variety of associative rings that satisfy the identity x 32 = 0, Electron. Res. Announc. Am. Math. Soc. 6 (2000), No. 7, 50–51.

    Article  MathSciNet  MATH  Google Scholar 

  84. A.V. Grishin, The variety of assiciative rings, which satisfy the identity x 32 = 0, is not Specht, in Editors: D. Krob et al., “Formal Power Series and Algebraic Combinatorics. Proceedings of the 12th International Conference, FPSAC‘, Moscow, Russia, June 26–30, 2000, Springer-Verlag, Berlin, 2000, 686–691.

    Google Scholar 

  85. C.K. Gupta, A.N. Krasilnikov, A non-finitely based system of polynomial identities which contains the identity x 6 = 0, Quart. J. Math. 53 (2002), 173–183.

    Article  MathSciNet  MATH  Google Scholar 

  86. C.K. Gupta, A.N. Krasilnikov, A simple example of a non-finitely based system of polynomial identities, Comm. Algebra 30 (2002), 4851–4866.

    Article  MathSciNet  MATH  Google Scholar 

  87. B.G. Gurevich, Foundations of the Theory of Algebraic Invariants, No-ordhoff, Groningen, 1964.

    Google Scholar 

  88. M. Hall, Jr., Combinatorial Theory, Blaisdell Publ. Comp., Waltham,Mass.-Toronto-London, 1967. Second Edition, Wiley-Interscience Series in Discrete Math., John Wiley & Sons, New York, 1986.

    Google Scholar 

  89. P. Halpin, Central and weak identities for matrices,Comm. Algebra 11 (1983), 2237–2248.

    Article  MathSciNet  MATH  Google Scholar 

  90. P. Halpin, Some Poincarü series related to identities of 2 x 2 matrices,Pacific J. Math. 107 (1983), 107–115.

    MathSciNet  MATH  Google Scholar 

  91. I.N. Herstein, Noncommutative Rings, Carus Math. Monographs 15 Wi-ley and Sons, Inc., New York, 1968.

    Google Scholar 

  92. G. Higman, On a conjecture of Nagata,Proc. Camb Philos. Soc. 52 (1956), 1–4.

    MathSciNet  MATH  Google Scholar 

  93. D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473–534; reprinted in “Gesammelte Abhandlungen, Band II Algebra, Invariantentheorie, Geometrie”, Zweite Auflage, Springer-Verlag, Berlin - Heidelberg - New York, 1970, 199–257.

    Article  MathSciNet  MATH  Google Scholar 

  94. D. Hilbert, Mathematische Probleme, Archiv f. Math. u. Phys. 1 (1901),44–63, 213–237; reprinted in “Gesammelte Abhandlungen, Band III Analysis, Grundlagen der Mathematik, Physik, Verschiedenes, Lebensgeschichte”, Zweite Auflage, Springer-Verlag, Berlin - Heidelberg - New York, 1970, 290–329.

    Google Scholar 

  95. N. Jacobson, PI-Algebras: An Introduction, Lecture Notes in Math. 441Springer-Verlag, Berlin - New York, 1975.

    Google Scholar 

  96. G. James, A. Kerber, The Representation Theory of the SymmetricGroup, Encyclopedia of Math. and Its Appl. 16 Addison-Wesley, Reading, Mass. 1981.

    Google Scholar 

  97. I. Kaplansky, Topological representations of algebras. I, Trans. Amer.Math. Soc. 68 (1950), 62–75.

    Article  MathSciNet  MATH  Google Scholar 

  98. I. Kaplansky,Problems in the theory of rings, Report of a Conference on Linear Algebras, June, 1956, in National Acad. of Sci. - National Research Council, Washington, Publ. 502 (1957), 1–3.

    Google Scholar 

  99. I. Kaplansky, Problems in the theory of rings revised, Amer. Math. Monthly 77 (1970), 445–454.

    Article  MathSciNet  MATH  Google Scholar 

  100. A.R. Kemer, Capelli identities and nilpotency of the radical of finitelygenerated PI-algebras (Russian), Dokl. Akad. Nauk SSSR 255 (1980), 793–797. Translation: Soy. Math. Dokl. 22 (1980), 750–753.

    MathSciNet  Google Scholar 

  101. A.R. Kemer, Varieties and Z 2 -graded algebras (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 48 (1984), 1042–1059. Translation: Math. USSR, Izv. 25 (1985), 359–374.

    MathSciNet  Google Scholar 

  102. A.R. Kemer, Finite basis property of identities of associative algebras (Russian), Algebra i Logika 26 (1987), 597–641. Translation: Algebra and Logic 26 (1988), 362–397.

    Article  MathSciNet  MATH  Google Scholar 

  103. A.R. Kemer, Representability of reduced-free algebras (Russian), Algebra i Logika 27 (1988), 274–294. Translation: Algebra and Logic 27 (1989), 167–184.

    Article  MathSciNet  Google Scholar 

  104. A.R. KemerIdeals of Identities of Associative Algebras, Translations of Math. Monographs 87 AMS, Providence, RI, 1991.

    Google Scholar 

  105. V.K. Kharchenko, A remark on central polynomials (Russian), Mat. Za-metki 26 (1979), 345–346. Translation: Math. Notes 26 (1979), p. 665.

    MathSciNet  Google Scholar 

  106. A.A. Kirillov, Certain division algebras over a field of rational functions (Russian), Funkts. Anal. Prilozh. 1 (1967), No.1, 101–102. Translation: Funct. Anal. Appl. 1 (1967), 87–88.

    MathSciNet  Google Scholar 

  107. P. KoshlukovFinitely based ideals of weak polynomial identities, Comm Algebra 26 (1998), 3335–3359.

    Article  MathSciNet  MATH  Google Scholar 

  108. B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitzki and cohomology theory, J. Math. Mech. 7 (1958), 237–264.

    MathSciNet  MATH  Google Scholar 

  109. D. Krakowski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438.

    MathSciNet  MATH  Google Scholar 

  110. G.R. Krause, T.H. Lenegan, Growth of Algebras and Gelfand-Kirillov Di-mension, Pitman Publ., London, 1985. Revised edition, Graduate Studies in Mathematics, 22 AMS, Providence, RI, 2000.

    Google Scholar 

  111. A.G. Kurosch, Ringtheoretische Probleme, die mit dem Burnsideschen Problem liber periodische Gruppen in Zusammenhang stehen (Russian, German summary), Bull. Acad. Sci. URSS Sür. Math. (Izv. Akad. Nauk SSSR) 5 (1941), 233–240.

    MathSciNet  MATH  Google Scholar 

  112. E.N. Kuzmin, On the Nagata-Higman theorem (Russian), in “Mathemat-ical Structures, Computational Mathematics, Mathematical Modelling. Proc. Dedicated to the 60th Birthday of Acad. L. Iliev”, Sofia, 1975, 101–107.

    Google Scholar 

  113. S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. Second Edition,1984.

    Google Scholar 

  114. V.N. Latyshev, Generalization of Hilbert’s theorem of the finiteness of bases (Russian), Sibirsk. Mat. Zh. 7 (1966), 1422–1424. Translation: Sib. Math. J. 7 (1966), 1112–1113.

    Google Scholar 

  115. V.N. Latyshev, On Regev’s theorem on identities in a tensor product of PI-algebras (Russian), Uspekhi Mat. Nauk 27 (1972), No. 4, 213–214.

    MATH  Google Scholar 

  116. V.N. Latyshev, A.L. Shmelkin, A certain problem of Kaplansky (Rus-sian), Algebra i Logika 8 (1969), 447–448. Translation: Algebra and Logic 8 (1969), p. 257.

    Google Scholar 

  117. L. Le Bruyn, The functional equation for Poincarü series of trace rings of generic 2 x 2 matrices, Israel J. Math. 52 (1985), 355–360.

    Article  MathSciNet  MATH  Google Scholar 

  118. L. Le Bruyn, Trace Rings of Generic 2 by 2 Matrices, Memoirs of AMS, 66 No. 363, Providence, R.I., 1987.

    Google Scholar 

  119. U. Leron, Multilinear identities of the matrix ring, Trans. Amer. Math.Soc. 183 (1973), 175–202.

    Article  MathSciNet  MATH  Google Scholar 

  120. U. Leron, A. VapnePolynomial identities of related rings,Israel J. Math.8 (1970), 127–137.

    MathSciNet  MATH  Google Scholar 

  121. I. LevitzkiOn a problem of A. Kurosch, Bull. Amer. Math. Soc. 52 (1946), 1033–1035.

    Article  MathSciNet  MATH  Google Scholar 

  122. I.V. Lvov, On the Shirshov height theorem (Russian), in “Fifth All-Union Symposium on the Theory of Rings, Algebras and Modules (Held at Novosibirsk, September, 21st-23rd, 1982)”. Abstracts of communications. Institute of Mathematics SO AN SSSR, Novosibirsk 1982, 89–90.

    Google Scholar 

  123. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press (Clarendon), Oxford, 1979. Second Edition, 1995.

    Google Scholar 

  124. A.I. Malcev, On algebras defined by identities (Russian), Mat. Sb. 26 (1950), 19–33.

    MathSciNet  Google Scholar 

  125. [ Yu.N. Maltsev, A basis for the identities of the algebra of upper triangular matrices (Russian), Algebra i Logika 10 (1971), 393–400. Translation: Algebra and Logic 10 (1971), 242–247.

    Google Scholar 

  126. V.T. Markov, The Gelfand-Kirillov dimension: nilpotency, representabil-ity, non-matrix varieties (Russian), Siberian School on Varieties of Algebraic Systems, Abstracts, Barnaul, 1988, 43–45. Zbl. 685.00002.

    Google Scholar 

  127. T. Molien, Ober die Invarianten der linearen Substitutionsgruppen, Sitz.König Preuss. Akad. Wiss. 52 (1897), 1152–1156.

    Google Scholar 

  128. M. Nagata, On the nilpotency of nil algebras, J. Math. Soc. Japan 4 (1953), 296–301.

    Article  MathSciNet  Google Scholar 

  129. M. Nagata, On the 14th problem of Hilbert, Amer. J. Math. 81 (1959),766–772.

    Article  MathSciNet  MATH  Google Scholar 

  130. K. Nakamoto, The structure of the invariant ring of two matrices ofdegree 3, J. Pure Appl. Algebra 166 (2002), No. 1–2, 125–148.

    Article  MathSciNet  MATH  Google Scholar 

  131. B.H. Neumann, Identical relations in groups. I, Math. Ann. 114 (1937),506–525.

    Article  Google Scholar 

  132. E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89–92; reprinted in “Gesammelte Abhandlungen. Collected Papers”, Springer-Verlag, Berlin - Heidelberg - New York - Tokyo, 1983, 181–184.

    Article  Google Scholar 

  133. S.V. Okhitin, On varieties defined by two-variable identities (Russian),Moskov. Gos. Univ. Moscow 1986 (manuscript deposited in VINITI 12.02.1986 No 1016-V), Ref. Zh. Mat. 6A366DEP/1986.

    Google Scholar 

  134. S.V. Okhitin, Central polynomials of an algebra of second order matrices(Russian), Vestnik Moskov. Univ. Ser. I, Mat. Mekh. (1988) No. 4, 61–63.Translation: Moscow Univ. Math. Bull. 43 (1988), No. 4, 49–51.

    Google Scholar 

  135. J. Olsson, A. Regev, On the T-ideal generated by a standard identity, Israel J. Math. 26 (1977), 97–104.

    Article  MathSciNet  MATH  Google Scholar 

  136. D.S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977.

    MATH  Google Scholar 

  137. V.M. Petrogradsky, On types of super exponential growth of identities in Lie PI-algebras (Russian), Fundam. Prikl. Mat. 1 (1995), 989–1007.

    MathSciNet  Google Scholar 

  138. V.M. Petrogradsky, Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions (Russian), Mat. Sb. 188 (1997), No. 6, 119–138. Translation: Sb. Math. 188 (1997), 913–931.

    Google Scholar 

  139. V.M. Petrogradsky, On the complexity functions for T -ideals of associa- tive algebras (Russian), Mat. Zametki 68 (2000), 887–897. Translation: Math. Notes 68 (2000), 751–759.

    Google Scholar 

  140. A.P. Popov, Identities of the tensor square of a Grassmann algebra (Rus-sian), Algebra i Logika 21 (1982), 442–471. Translation: Algebra and Logic 21 (1982), 296–316.

    Google Scholar 

  141. C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Rend.Cl. Sci. Fis. Mat. Natur. 8 (1967), 239–255.

    MathSciNet  MATH  Google Scholar 

  142. C. Procesi, Rings with Polynomial Identities, Marcel Dekker, New York, 1973.

    Google Scholar 

  143. C. Procesi, The invariant theory of n x n matrices, Adv. Math. 19 (1976), 306–381.

    MathSciNet  MATH  Google Scholar 

  144. C. Procesi, Computing with 2 x 2 matrices, J. Algebra 87 (1984), 342–359.

    Article  MathSciNet  MATH  Google Scholar 

  145. Yu.P. Razmyslov, Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (Russian), Algebra i Logika 12 (1973), 83–113. Translation: Algebra and Logic 12 (1973) 47–63.

    Article  MathSciNet  Google Scholar 

  146. Yu.P. Razmyslov, On a problem of Kaplansky (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 483–501. Translation: Math. USSR, Izv. 7 (1973), 479–496.

    MathSciNet  Google Scholar 

  147. Yu.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 723–756. Translation: Math. USSR, Izv. 8 (1974), 727–760.

    MathSciNet  Google Scholar 

  148. Yu.P. Razmyslov, The Jacobson radical in PI-algebras (Russian), Algebra i Logika 13 (1974), 337–360. Translation: Algebra and Logic 13 (1974), 192–204.

    MathSciNet  Google Scholar 

  149. Yu.P. Razmyslov, Identities of Algebras and Their Representations (Rus-sian), “Sovremennaya Algebra”, “Nauka”, Moscow, 1989. Translation: Translations of Math. Monographs 138 AMS, Providence, R.I., 1994.

    Google Scholar 

  150. A. Regev, Existence of identities in A ⊗ B, Israel J. Math. 11 (1972), 131–152.

    Article  MathSciNet  MATH  Google Scholar 

  151. A. Regev, The representation of S n and explicit identities for P.I. algebras, J. Algebra 51 (1978), 25–40.

    Article  MathSciNet  MATH  Google Scholar 

  152. A. Regev, Algebras satisfying a Capelli identity, Israel J. Math. 33 (1979), 149–154.

    Article  MathSciNet  MATH  Google Scholar 

  153. A. Regev, The Kronecker product of S n - characters and an A ® B theorem for Capelli identities, J. Algebra 66 (1980), 505–510.

    Article  MathSciNet  MATH  Google Scholar 

  154. A. Regev, On the codimensions of matrix algebras, in “Algebra — Some Current Trends (Varna, 1986)”, Lect. Notes in Math. 1352 Springer-Verlag, Berlin-New York, 162–172, 1988.

    Google Scholar 

  155. A. Regev, Tensor products of matrix algebras over the Grassmann algebra, J. Algebra 133 (1990), 512–526.

    Article  MathSciNet  MATH  Google Scholar 

  156. A. Regev, Asymptotics of codimensions of some P.I. algebras, in “Trends in Ring Theory” (Proc. Conf. Miskolc, 1996; Eds. V. Dlab and L. Márki), CMS Conference Proceedings 22 AMS, Providence, 1998,159–172.

    Google Scholar 

  157. S. Rosset, A new proof of the Amitsur-Levitzki identity, Israel J. Math. 23 (1976), 187–188.

    Article  MathSciNet  MATH  Google Scholar 

  158. L.H. Rowen, Polynomial Identities of Ring Theory,Acad. Press, 1980.

    Google Scholar 

  159. L.H. Rowen, Ring Theory, vol. 1, 2, Academic Press, Inc., Boston, MA,1988. Student Edition, Academic Press, Inc., Boston, MA, 1991.

    Google Scholar 

  160. V.V. Shchigolev, Examples of infinitely based T-ideals (Russian, English summary), Fundam. Prikl. Mat. 5 (1999), No. 1, 307–312.

    MathSciNet  MATH  Google Scholar 

  161. I.P. Shestakov, N. Zhukavets, On associative algebras satisfying the iden- tity x 5 = 0, preprint.

    Google Scholar 

  162. A.I. Shirshov, On rings with identity relations (Russian), Mat. Sb. 41 (1957), 277–283.

    MathSciNet  Google Scholar 

  163. K.S. Sibirskii, Algebraic invariants for a set of matrices (Russian), Sib.Mat. Zh. 9 (1968), No. 1, 152–164. Translation: Siber. Math. J. 9 (1968), 115–124.

    Article  MathSciNet  Google Scholar 

  164. W. Specht, Gesetze in Ringen. I, Math. Z. 52 (1950), 557–589.

    Article  MathSciNet  MATH  Google Scholar 

  165. R.P. Stanley, Combinatorics and invariant theory, in “Relations between Combinatorics and Other Parts of Mathematics” Ed. D. K. RayChaudhuri, Proc. Sympos. Pure Math., 34, AMS, Providence, RI, 1979, 345–355.

    Google Scholar 

  166. R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Math. 41, Birkhäuser Boston, Boston, MA, 1983.

    Google Scholar 

  167. R.G. Swan, An application of graph theory to algebra, Proc. Amer. Math.Soc. 14 (1963), 367–373. Correction: 21 (1969), 379–380.

    Google Scholar 

  168. J. Szigeti, Z. Tuza, G. Révész, Eulerian polynomial identities on matrix rings, J. Algebra 161 (1993), 90–101.

    Article  MathSciNet  MATH  Google Scholar 

  169. Y. Teranishi, The ring of invariants of matrices, Nagoya Math. J. 104 (1986), 149–161.

    MathSciNet  MATH  Google Scholar 

  170. Y. Teranishi, Linear diophantine equations and invariant theory of ma- trices,“Commut Algebra and Combinatorics (Kyoto, 1985)”, Adv. Stud. Pure Math. 11, North-Holland, Amsterdam - New York, 1987, 259–275.

    Google Scholar 

  171. Y. Teranishi, The Hilbert series of matrix concominants, Nagoya Math. J. 111 (1988), 143–156.

    MathSciNet  MATH  Google Scholar 

  172. V.A. Ufnarovski, An independence theorem and its consequences (Russian), Mat. Sb. 128 (1985), 124–132. Translation: Math. USSR Sb. 56 (1985), 121–129.

    MathSciNet  Google Scholar 

  173. V.A. Ufnarovski, Combinatorial and asymptotic methods in algebra,in A.I. Kostrikin, I R Shafarevich (Eds.), “Algebra VI”, Encyclopedia of Math. Sciences 57, Springer-Verlag, 1995, 1–196.

    Google Scholar 

  174. M. Van den BerghTrace rings are Cohen-Macaulay, J. Amer. Math. Soc. 2 (1989), 775–799.

    Article  MathSciNet  MATH  Google Scholar 

  175. M. Van den BerghExplicit rational forms for the Poincarü series of the trace rings of generic matrices, Israel J. Math. 73 (1991), 17–31.

    MATH  Google Scholar 

  176. M.R. Vaughan-LeeAn algorithm for computing graded algebras, J. Sym-bolic Comput. 16 (1993), 345–354.

    Article  MathSciNet  MATH  Google Scholar 

  177. U. VishnePolynomial identities of M2(G), Comm Algebra 30 (2002),443–454.

    Article  MathSciNet  MATH  Google Scholar 

  178. W. WagnerOber die Grundlagen der projektiven Geometrie und allge-meine Zahlsysteme, Math. Z. 113 (1937), 528–567.

    Google Scholar 

  179. H. Weyl, Zur Darstellungstheorie und Invariantenabzählung der projek-tiven,der Komplex-und der Drehungsgruppe, Acta Math. 48 (1926), 255–278; reprinted in “Gesammelte Abhandlungen”, Band III, Springer - Verlag, Berlin-Heidelberg - New York, 1968, 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  180. H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ. Press, Princeton, N.J., 1946, New Edition, 1997.

    Google Scholar 

  181. K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov, A.I. Shirshov, Rings That Are Nearly Associative (Russian), “Nauka”, Moscow, 1978. Translation: Academic Press, New York, 1982.

    MATH  Google Scholar 

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  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

    Vesselin Drensky

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Edward Formanek

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Drensky, V., Formanek, E. (2004). Bibliography. In: Polynomial Identity Rings. Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7934-7_10

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