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1567131
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Book Information:
Author:
Donald S. Passman
Title:
The algebraic structure of group rings
Additional book information:
John Wiley & Sons, New York, London, Sydney, Toronto, 1977, xiv + 720 pp., $34.95.
S. A. Amitsur, The radical of field extensions, Bull. Res. Council Israel Sect. F 7F (1957/58), 1–10. MR 103912
S. A. Amitsur, On the semi-simplicity of group algebras, Michigan Math. J. 6 (1959), 251–253. MR 108540
S. A. Amitsur, Groups with representations of bounded degree. II, Illinois J. Math. 5 (1961), 198–205. MR 122893
George M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971), 459–469. MR 280489, DOI 10.1090/S0002-9947-1971-0280489-8
S. D. Berman, Group algebras of countable abelian $p$-groups, Publ. Math. Debrecen 14 (1967), 365–405 (Russian). MR 225897
A. A. Bovdi, Group rings of torsion-free groups, Sibirsk. Mat. Ž. 1 (1960), 555–558 (Russian). MR 0130919
R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556–590. MR 4042, DOI 10.2307/1968918
Kenneth A. Brown, On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), no. 3, 251–256. MR 414616, DOI 10.1112/blms/8.3.251
Ian G. Connell, On the group ring, Canadian J. Math. 15 (1963), 650–685. MR 153705, DOI 10.4153/CJM-1963-067-0
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
Everett C. Dade, Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345–348 (French). MR 280610, DOI 10.1007/BF01109886
Daniel Farkas, Self-injective group algebras, J. Algebra 25 (1973), 313–315. MR 313300, DOI 10.1016/0021-8693(73)90049-5
Daniel R. Farkas and D. S. Passman, Primitive Noetherian group rings, Comm. Algebra 6 (1978), no. 3, 301–315. MR 469961, DOI 10.1080/00927877808822247
Daniel R. Farkas and Robert L. Snider, Group algebras whose simple modules are injective, Trans. Amer. Math. Soc. 194 (1974), 241–248. MR 357475, DOI 10.1090/S0002-9947-1974-0357475-5
Daniel R. Farkas and Robert L. Snider, $K_{0}$ and Noetherian group rings, J. Algebra 42 (1976), no. 1, 192–198. MR 422327, DOI 10.1016/0021-8693(76)90036-3
Edward Formanek, Idempotents in Noetherian group rings, Canadian J. Math. 25 (1973), 366–369. MR 316494, DOI 10.4153/CJM-1973-037-6
Edward Formanek, Group rings of free products are primitive, J. Algebra 26 (1973), 508–511. MR 321964, DOI 10.1016/0021-8693(73)90011-2
Edward Formanek, The zero divisor question for supersolvable groups, Bull. Austral. Math. Soc. 9 (1973), 69–71. MR 325670, DOI 10.1017/S000497270004288X
Edward Formanek and Robert L. Snider, Primitive group rings, Proc. Amer. Math. Soc. 36 (1972), 357–360. MR 308178, DOI 10.1090/S0002-9939-1972-0308178-8
Enzo R. Gentile, A note on injective group rings, Proc. Amer. Math. Soc. 23 (1969), 431–432. MR 245693, DOI 10.1090/S0002-9939-1969-0245693-X
Alfred W. Goldie, The structure of Noetherian rings, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 213–321. MR 0393118
J.-M. Goursaud, Sur les anneaux de groupes semi-parfaits, Canadian J. Math. 25 (1973), 922–928 (French). MR 338054, DOI 10.4153/CJM-1973-098-1
Jean-Marie Goursaud and Jacques Valette, Sur l’enveloppe injective des anneaux de groupes réguliers, Bull. Soc. Math. France 103 (1975), no. 1, 91–102 (French). MR 379569
P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436. MR 72873, DOI 10.1112/plms/s3-4.1.419
P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622. MR 110750, DOI 10.1112/plms/s3-9.4.595
B. Hartley, The residual nilpotence of wreath products, Proc. London Math. Soc. (3) 20 (1970), 365–392. MR 258966, DOI 10.1112/plms/s3-20.3.365
B. Hartley, Injective modules over group rings, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 109, 1–29. MR 435127, DOI 10.1093/qmath/28.1.1
K. A. Hirsch, On infinite soluble groups. III, Proc. London Math. Soc. (2) 49 (1946), 184–194. MR 17281, DOI 10.1112/plms/s2-49.3.184
I. M. Isaacs and D. S. Passman, Groups with representations of bounded degree, Canadian J. Math. 16 (1964), 299–309. MR 167539, DOI 10.4153/CJM-1964-029-5
I. M. Isaacs and D. S. Passman, A characterization of groups in terms of the degrees of their characters, Pacific J. Math. 15 (1965), 877–903. MR 191972
Gerald J. Janusz, Algebraic number fields, Pure and Applied Mathematics, Vol. 55, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0366864
Arun Vinayak Jategaonkar, Integral group rings of polycyclic-by-finite groups, J. Pure Appl. Algebra 4 (1974), 337–343. MR 344345, DOI 10.1016/0022-4049(74)90013-9
S. A. Jennings, The structure of the group ring of a $p$-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. MR 4626, DOI 10.1090/S0002-9947-1941-0004626-6
S. A. Jennings, The group ring of a class of infinite nilpotent groups, Canadian J. Math. 7 (1955), 169–187. MR 68540, DOI 10.4153/CJM-1955-022-5
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
Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496
Jacques Lewin, A note on zero divisors in group-rings, Proc. Amer. Math. Soc. 31 (1972), 357–359. MR 292957, DOI 10.1090/S0002-9939-1972-0292957-X
A. I. Mal′cev, On the embedding of group algebras in division algebras, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 1499–1501 (Russian). MR 0025457
A. I. Mal′cev, On some classes of infinite soluble groups, Mat. Sbornik N.S. 28(70) (1951), 567–588 (Russian). MR 0043088
40. M. Malliavin, Sur les anneaux de groupes FP self-infectifs, C. R. Acad. Sci. Paris 273 (1971), 88-91. MR 44 #5386.
Warren May, Commutative group algebras, Trans. Amer. Math. Soc. 136 (1969), 139–149. MR 233903, DOI 10.1090/S0002-9947-1969-0233903-9
Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. MR 32593, DOI 10.1090/S0002-9947-1949-0032593-5
Masaru Osima, Notes on blocks of group characters, Math. J. Okayama Univ. 4 (1955), 175–188. MR 78364
D. S. Passman, Nil ideals in group rings, Michigan Math. J. 9 (1962), 375–384. MR 144930
Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
D. S. Passman, Primitive group rings, Pacific J. Math. 47 (1973), 499–506. MR 332942
D. S. Passman, Group rings satisfying a polynomial identity, J. Algebra 20 (1972), 103–117. MR 285631, DOI 10.1016/0021-8693(72)90091-9
D. S. Passman, Group rings satisfying a polynomial identity. III, Proc. Amer. Math. Soc. 31 (1972), 87–90. MR 289671, DOI 10.1090/S0002-9939-1972-0289671-3
D. S. Passman, A new radical for group rings?, J. Algebra 28 (1974), 556–572. MR 427363, DOI 10.1016/0021-8693(74)90060-X
D. S. Passman, Advances in group rings, Israel J. Math. 19 (1974), 67–107. MR 357477, DOI 10.1007/BF02756627
D. S. Passman, What is a group ring?, Amer. Math. Monthly 83 (1976), no. 3, 173–185. MR 390033, DOI 10.2307/2977018
D. S. Passman, The Jacobson radical of a group ring of a locally solvable group, Proc. London Math. Soc. (3) 38 (1979), no. 1, 169–192. MR 520978, DOI 10.1112/plms/s3-38.1.169
Sam Perlis and Gordon L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420–426. MR 34758, DOI 10.1090/S0002-9947-1950-0034758-3
Edward C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180–183. MR 111765, DOI 10.1090/S0002-9939-1960-0111765-5
Guy Renault, Sur les anneaux de groupes, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A84–A87 (French). MR 288189
J. E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307–328. MR 332944, DOI 10.1016/0022-4049(73)90034-0
J. E. Roseblade and P. F. Smith, A note on hypercentral group rings, J. London Math. Soc. (2) 13 (1976), no. 1, 183–190. MR 399146, DOI 10.1112/jlms/s2-13.1.183
Alan Rosenberg, On the primitivity of the group algebra, Canadian J. Math. 23 (1971), 536–540. MR 286836, DOI 10.4153/CJM-1971-057-8
Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006
Martha Smith, Group algebras, J. Algebra 18 (1971), 477–499. MR 277625, DOI 10.1016/0021-8693(71)90078-0
P. F. Smith, On the intersection theorem, Proc. London Math. Soc. (3) 21 (1970), 385–398. MR 294382, DOI 10.1112/plms/s3-21.3.385
P. F. Smith, Localization in group rings, Proc. London Math. Soc. (3) 22 (1971), 69–90. MR 294384, DOI 10.1112/plms/s3-22.1.69
Jacques Valette, Anneaux de groupes semi-parfaits, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1219–A1222 (French). MR 308275
Orlando E. Villamayor, On the semisimplicity of group algebras, Proc. Amer. Math. Soc. 9 (1958), 621–627. MR 98769, DOI 10.1090/S0002-9939-1958-0098769-7
Orlando E. Villamayor, On weak dimension of algebras, Pacific J. Math. 9 (1959), 941–951. MR 108527
R. Walker, Local rings and normalizing sets of elements, Proc. London Math. Soc. (3) 24 (1972), 27–45. MR 294399, DOI 10.1112/plms/s3-24.1.27
68. A. Whitcomb, The group ring problem, Ph.D. Thesis, Univ. of Chicago, 1967.
S. M. Woods, On perfect group rings, Proc. Amer. Math. Soc. 27 (1971), 49–52. MR 271247, DOI 10.1090/S0002-9939-1971-0271247-4
A. E. Zalesskiĭ, The group algebras of solvable groups, Vescī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 1970 (1970), no. 2, 13–21 (Russian). MR 0289675
71. A. E. Zalesskiĭ, On a problem of Kaplansky, Soviet Math. 13 (1972), 449-452. MR 45 #6947.
72. A. E. Zalesskiĭ, On the semisimplicity of a modular group algebra of a solvable group, Soviet Math. 14 (1973), 101-105.
A. E. Zalesskiĭ and A. V. Mihalev, Group rings, Current problems in mathematics, Vol. 2 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1973, pp. 5–118. (errata insert) (Russian). MR 0414623
- 1.
- S. A. Amitsur, The radical of field extensions, Bull. Res. Council Israel, 7F (1957), 1-10. MR 21 #2675. MR 0103912
- 2.
- S. A. Amitsur, On the semi-simplicity of group algebras, Michigan Math. J. 6 (1959), 251-253. MR 21 #7256. MR 108540
- 3.
- S. A. Amitsur, Groups with representations of bounded degree. II, Illinois J. Math. 5 (1961), 198-205. MR 23 # A225. MR 122893
- 4.
- G. M. Bergman, The logarithmic limit-set of an algebraic variety. Trans. Amer. Math. Soc. 157 (1971), 459-469. MR 43 #6209. MR 280489
- 5.
- S. D. Berman, Group algebras of countable abelian p-groups, Publ. Math. Debrecen 14 (1967), 365-405. (Russian) MR 37 # 1488. MR 225897
- 6.
- A. A. Bovdi, Group rings of torsion free groups, Sibirsk. Mat. Z. 1 (1960), 555-558. (Russian) MR 24 #A773. MR 130919
- 7.
- R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556-590. MR 2, p. 309. MR 4042
- 8.
- K. A. Brown, On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), 251-256. MR 54 #2716. MR 414616
- 9.
- I. G. Connell, On the group ring, Canad. J. Math. 15 (1963), 650-685. MR 27 #3666. MR 153705
- 10.
- C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience, New York, 1962. MR 26 #2519. MR 144979
- 11.
- E. C. Dade, Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345-348. MR 43 #6329. MR 280610
- 12.
- D. R. Farkas, Self-injective group algebras, J. Algebra 25 (1973), 313-315. MR 47 # 1855. MR 313300
- 13.
- D. R. Farkas and D. S. Passman, Primitive Noetherian group rings, Comm. Algebra 6 (1978), 301-315. MR 469961
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- D. R. Farkas and R. L. Snider, Group algebras whose simple modules are infective, Trans. Amer. Math. Soc. 194 (1974), 241-248. MR 50 #9943. MR 357475
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- E. R. Gentile, A note on injective group rings, Proc. Amer. Math. Soc. 23 (1969), 431-432. MR 39 #6999. MR 245693
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- J. M. Goursaud, Sur les anneaux de groupes semi-parfaits, Canad. J. Math. 25 (1973), 922-928. MR 49 #2821. MR 338054
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- J. M. Goursaud and J. Valette, Sur l'envelope infective des anneaux de groupes reguliers, Bull. Math. Soc. France 103 (1975), 91-102. MR 52 #474. MR 379569
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- P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595-622. MR 22 #1618. MR 110750
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- B. Hartley, The residual nilpotence of wreath products, Proc. London Math. Soc. (3) 20 (1970), 365-392. MR 41 #3611. MR 258966
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- B. Hartley, Injective modules over group rings, Quart. J. Math. (2) 28 (1977), 1-29. MR 435127
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- K. A. Hirsch, On infinite soluble groups. III, Proc. London Math. Soc. (2) 49 (1946), 184-194. MR 8, p. 132. MR 17281
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- I. M. Isaacs and D. S. Passman, Groups with representations of bounded degree, Canad. J. Math. 16 (1964), 299-309. MR 29 #4811. MR 167539
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- I. M. Isaacs and D. S. Passman, A characterization of groups in terms of the degrees of their characters, Pacific J. Math. 15 (1965), 877-903. MR 33 #199. MR 191972
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- C. J. Janusz, Algebraic number fields, Academic Press, New York, 1973. MR 366864
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- A. V. Jategaonkar, Integral group rings of polycyclic-by-finite groups, J. Pure Appl. Algebra 4 (1974), 337-343. MR 49 #9084. MR 344345
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- S. A. Jennings, The structure of the group ring of a p-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175-185. MR 3, p. 34. MR 4626
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- S. A. Jennings, The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169-187. MR 16, p. 899. MR 68540
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- I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580. MR 10, p. 7. MR 25451
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- M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. École Norm. Sup. (3) 71 (1954), 101-190. MR 19, p. 529. MR 88496
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- J. Lewin, A note on zero divisors in group rings, Proc. Amer. Math. Soc. 31 (1972), 357-359. MR 45 #2038. MR 292957
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- A. I. Malcev, On embedding of group algebras in a division algebra, Dokl. Akad. Nauk SSSR 60 (1948), 1499-1501. (Russian) MR 10, p. 8. MR 25457
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- A. I. Malcev, On some classes of infinite solvable groups, Mat. Sb. 28 (1951), 567-588. (Russian) MR 13, p. 203. MR 43088
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- M. Malliavin, Sur les anneaux de groupes FP self-infectifs, C. R. Acad. Sci. Paris 273 (1971), 88-91. MR 44 #5386.
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- A. O. Morris, Review of infinite group rings, by Donald S. Passman, Marcel Dekker, New York, 1971. MR 47 #3500. MR 314951
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- D. S. Passman, Nil ideals in group rings, Michigan Math. J. 9 (1962), 375-384. MR 26 #2470. MR 144930
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- D. S. Passman, Infinite group rings, Pure and Appl. Math., No. 6, Marcel Dekker, New York, 1971. MR 47 #3500. MR 314951
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- D. S. Passman, Primitive group rings, Pacific J. Math. 47 (1973), 499-506. MR 48 # 11267. MR 332942
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Review Information:
Reviewer:
D. A. R. Wallace
Journal:
Bull. Amer. Math. Soc.
1 (1979), 402-410
DOI:
https://doi.org/10.1090/S0273-0979-1979-14616-0