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Book Information:
Authors:
Alexei Kanel-Belov and
Louis Halle Rowen
Title:
Computational aspects of polynomial identities
Additional book information:
Research Notes in Mathematics, vol. 9, A~K Peters, Ltd.,
Wellesley, MA,
2005,
xxi + 378 pp.,
ISBN 1-56881-163-2,
US$69.00$
S. A. Amitsur, The $T$-ideals of the free ring, J. London Math. Soc. 30 (1955), 470–475. MR 71408, DOI 10.1112/jlms/s1-30.4.470
A. S. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449–463. MR 36751, DOI 10.1090/S0002-9939-1950-0036751-9
A. Z. Anan′in, Representability of Noetherian finitely generated algebras, Arch. Math. (Basel) 59 (1992), no. 1, 1–5. MR 1166010, DOI 10.1007/BF01199007
A. Ya. Belov, On non-Specht varieties, Fundam. Prikl. Mat. 5 (1999), no. 1, 47–66 (Russian, with English and Russian summaries). MR 1799544
Amiram Braun, The nilpotency of the radical in a finitely generated PI ring, J. Algebra 89 (1984), no. 2, 375–396. MR 751151, DOI 10.1016/0021-8693(84)90224-2
[6] M. Dehn, Über die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Ann. 85 (1922), 184-193.
Vesselin Drensky and Edward Formanek, Polynomial identity rings, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004. MR 2064082, DOI 10.1007/978-3-0348-7934-7
Edward Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129–132. MR 302689, DOI 10.1016/0021-8693(72)90050-6
A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221–243. MR 1680198, DOI 10.1006/aima.1998.1790
[10] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Amer. Math. Soc., 2005.
A. V. Grishin, A variety of associative rings is not Spechtian, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 157–158 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 1025–1026. MR 1741669, DOI 10.1070/rm1999v054n05ABEH000209
Marshall Hall, Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277. MR 8892, DOI 10.1090/S0002-9947-1943-0008892-4
Irving Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575–580. MR 25451, DOI 10.1090/S0002-9904-1948-09049-8
A. R. Kemer, Capelli identities and nilpotency of the radical of finitely generated PI-algebra, Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 793–797 (Russian). MR 600746
A. R. Kemer, Solution of the problem as to whether associative algebras have a finite basis of identities, Dokl. Akad. Nauk SSSR 298 (1988), no. 2, 273–277 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 1, 60–64. MR 937115
Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR 1108620, DOI 10.1090/mmono/087
A. R. Kemer, The standard identity in characteristic $p$: a conjecture of I. B. Volichenko, Israel J. Math. 81 (1993), no. 3, 343–355. MR 1231198, DOI 10.1007/BF02764837
Bertram Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. Math. Mech. 7 (1958), 237–264. MR 0092755, DOI 10.1512/iumj.1958.7.57019
V. N. Latyšev, On Regev’s theorem on identities in a tensor product of $\textrm {PI}$-algebras, Uspehi Mat. Nauk 27 (1972), no. 4(166), 213–214 (Russian). MR 0393114
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
Ju. P. Razmyslov, A certain problem of Kaplansky, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 483–501 (Russian). MR 0338063
Ju. P. Razmyslov, The Jacobson Radical in PI-algebras, Algebra i Logika 13 (1974), 337–360, 365 (Russian). MR 0419515
Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756 (Russian). MR 0506414
Amitai Regev, Existence of identities in $A\otimes B$, Israel J. Math. 11 (1972), 131–152. MR 314893, DOI 10.1007/BF02762615
Amitai Regev, Asymptotics of codimensions of some P.I. algebras, Trends in ring theory (Miskolc, 1996) CMS Conf. Proc., vol. 22, Amer. Math. Soc., Providence, RI, 1998, pp. 159–172. MR 1491923
William Schelter, Non-commutative affine P.I. rings are catenary, J. Algebra 51 (1978), no. 1, 12–18. MR 485980, DOI 10.1016/0021-8693(78)90131-X
V. V. Shchigolev, Examples of infinitely based $T$-ideals, Fundam. Prikl. Mat. 5 (1999), no. 1, 307–312 (Russian, with English and Russian summaries). MR 1799533
A. I. Širšov, On some non-associative null-rings and algebraic algebras, Mat. Sb. N.S. 41(83) (1957), 381–394 (Russian). MR 0089841
A. I. Širšov, On rings with identity relations, Mat. Sb. N.S. 43(85) (1957), 277–283 (Russian). MR 0095192
Lance W. Small, Prime ideals in Noetherian $\textrm {PI}$-rings, Bull. Amer. Math. Soc. 79 (1973), 421–422. MR 313307, DOI 10.1090/S0002-9904-1973-13196-9
Wilhelm Specht, Gesetze in Ringen. I, Math. Z. 52 (1950), 557–589 (German). MR 35274, DOI 10.1007/BF02230710
[32] W. Wagner, Über die Grundlagen der projektiven Geometrie und allgemeine Zahlststeme, Math. Z. 113 (1937), 528-567.
- [1]
- S. A. Amitsur, The T-ideals of the free ring, J. London Math. Soc. 30 (1955), 470-475. MR 0071408 (17:122c)
- [2]
- S. A. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463. MR 0036751 (12:155d).
- [3]
- A. Z. Anan'in, The representability of finitely generated algebras with chain condition, Arch. Math. 59 (1992), 1-5. MR 1166010 (93g:16036)
- [4]
- A. Ya. Belov, On non-Specht varieties, Fundam. Prikl. Mat. 5 (1999), 47-66 (Russian). MR 1799544 (2001k:16040)
- [5]
- A. Braun, The nilpotency of the radical in a finitely generated PI-ring, J. Algebra 89 (1984), 375-396. MR 0751151 (85m:16007)
- [6]
- M. Dehn, Über die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Ann. 85 (1922), 184-193.
- [7]
- V. Drensky and E. Formanek, Polynomial Identity Rings, Birkhäuser, 2004. MR 2064082 (2005c:16040)
- [8]
- E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129-132. MR 0302689 (46:1833)
- [9]
- A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), 221-243. MR 1680198 (2000a:16048)
- [10]
- A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Amer. Math. Soc., 2005.
- [11]
- A. V. Grishin, A variety of associative rings is not Spechtian, Uspekhi Mat. Nauk 54 (1999), 157-158 (Russian). Translation: Russian Math. Surveys 54 (1999), 1025-1026. MR 1741669 (2000m:16035)
- [12]
- M. Hall, Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229-277. MR 0008892 (5:72c)
- [13]
- I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580. MR 0025451 (10:7a)
- [14]
- A. R. Kemer, Capelli identities and nilpotency of the radical in finitely generated PI-algebras, Dokl. Akad. Nauk SSSR 255 (1980), 793-797 (Russian). Translation: Soviet Math. Dokl. 22 (1980). MR 0600746 (82c:16019)
- [15]
- A. R. Kemer, Solution of the problem as to whether associative algebras have a finite basis of identities, Dokl. Akad. Nauk SSSR 298 (1988), 273-277. MR 0937115 (89d:16023)
- [16]
- A. R. Kemer, Ideals of Identities of Associative Algebras, Amer. Math. Soc., 1991. MR 1108620 (92f:16031)
- [17]
- A. R. Kemer, The standard identity in characteristic : a conjecture of I. B. Volichenko, Israel J. Math. 81 (1993), 343-355. MR 1231198 (94f:16040)
- [18]
- B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. Math. Mech. 7 (1958), 237-264. MR 0092755 (19:1153e)
- [19]
- V. N. Latyshev, On Regev's theorem on identities in a tensor product of PI-algebras, Uspekhi Mat. Nauk 27 (1972), 213-214 (Russian). MR 0393114 (52:13924)
- [20]
- C. Procesi, The invariant theory of matrices, Adv. in Math. 19 (1976), 306-381. MR 0419491 (54:7512)
- [21]
- Y. P. Razmyslov, On a certain problem of Kaplansky, Izv. Akad. Nauk SSSR 37 (1973), 483-501 (Russian). Translation: Math. USSR-Izv. 7 (1973), 479-496. MR 0338063 (49:2830)
- [22]
- Y. P. Razmyslov, The Jacobson radical in PI-algebras, Alg. i Logika 13 (1974), 337-360 (Russian). Translation: Algebra and Logic 13 (1974), 192-204. MR 0419515 (54:7536)
- [23]
- Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR 38 (1974), 723-756 (Russian). Translation: Math. USSR-Izv. 8 (1974), 727-760. MR 0506414 (58:22158)
- [24]
- A. Regev, Existence of identities in , Israel J. Math. 11 (1972), 131-152. MR 0314893 (47:3442)
- [25]
- A. Regev, Asymptotics of codimensions of some P.I. algebras, pp. 159-172 in Trends in Ring Theory (Miskolc, 1996), V. Dlab and L. Márki, Editors, Canadian Math. Soc. Conference Proceedings Vol. 22, Amer. Math. Soc., 1998. MR 1491923 (98k:16033)
- [26]
- W. Schelter, Noncommutative affine PI rings are catenary, J. Algebra 51 (1978), 12-18. MR 0485980 (58:5772)
- [27]
- V. V. Shchigolev, Examples of infinitely based T-ideals, Fundam. Prikl. Mat 5 (1999), 307-321 (Russian). MR 1799533 (2001k:16044)
- [28]
- A. I. Shirshov, On some non-associative nil-rings and algebraic algebras, Mat. Sb. 41(83) (1957), 381-394 (Russian). MR 0089841 (19:727h)
- [29]
- A. I. Shirshov, On rings with identity relations, Math. Sb. 43(85) (1957), 277-283 (Russian). MR 0095192 (20:1698)
- [30]
- L. W. Small, Prime ideals in Noetherian PI-rings, Bull. Amer. Math. Soc. 79 (1973), 421-422. MR 0313307 (47:1862)
- [31]
- W. Specht, Gesetze in Ringen. I, Math. Z. 52 (1950), 557-589. MR 0035274 (11:711i)
- [32]
- W. Wagner, Über die Grundlagen der projektiven Geometrie und allgemeine Zahlststeme, Math. Z. 113 (1937), 528-567.
Review Information:
Reviewer:
Edward Formanek
Affiliation:
The Pennsylvania State University
Email:
formanek@math.psu.edu
Journal:
Bull. Amer. Math. Soc.
43 (2006), 579-584
Published electronically:
April 20, 2006
Review copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.