Book Review
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MathSciNet review:
1567914
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Book Information:
Author:
D. S. Passman
Title:
Infinite crossed products
Additional book information:
Academic Press, New York, 460 pp., $84.50. ISBN 0-12-546390-1.
A. Adrian Albert and Helmut Hasse, A determination of all normal division algebras over an algebraic number field, Trans. Amer. Math. Soc. 34 (1932), no. 3, 722–726. MR 1501659, DOI 10.1090/S0002-9947-1932-1501659-X
S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420. MR 318216, DOI 10.1007/BF02764632
Gorô Azumaya, New foundation of the theory of simple rings, Proc. Japan Acad. 22 (1946), no. 11, 325–332. MR 42387
A. A. Bovdi, Crossed products of a semigroup and a ring, Dokl. Akad. Nauk SSSR 137 (1961), 1267–1269 (Russian). MR 0132766
[BHN] R. Brauer, H. Hasse, and E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, J. für Math. 167 (1932), 399-404.
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
[D1] L. E. Dickson, Abstract, Bull. Amer. Math. Soc. (N.S.) 12 (1906), 441-442.
L. E. Dickson, Linear associative algebras and abelian equations, Trans. Amer. Math. Soc. 15 (1914), no. 1, 31–46. MR 1500963, DOI 10.1090/S0002-9947-1914-1500963-8
George A. Elliott, Automorphisms determined by multipliers on ideals of a $C^*$-algebra, J. Functional Analysis 23 (1976), no. 1, 1–10. MR 0440372, DOI 10.1016/0022-1236(76)90054-9
Joe W. Fisher and Susan Montgomery, Semiprime skew group rings, J. Algebra 52 (1978), no. 1, 241–247. MR 480616, DOI 10.1016/0021-8693(78)90272-7
Joe W. Fisher and James Osterburg, Some results on rings with finite group actions, Ring theory (Proc. Conf., Ohio Univ., Athens, Ohio, 1976) Lecture Notes in Pure and Appl. Math., Vol. 25, Dekker, New York, 1977, pp. 95–111. MR 0442022
N. Jacobson, The fundamental theorem of the Galois theory for quasi-fields, Ann. of Math. (2) 41 (1940), 1–7. MR 1219, DOI 10.2307/1968817
Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601
V. K. Harčenko, The Galois theory of semiprime rings, Algebra i Logika 16 (1977), no. 3, 313–363 (Russian). MR 0573067
V. K. Harčenko, Algebras of invariants of free algebras, Algebra i Logika 17 (1978), no. 4, 478–487, 491 (Russian). MR 538309
V. K. Kharchenko, Automorphisms and derivations of associative rings, Mathematics and its Applications (Soviet Series), vol. 69, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian by L. Yuzina. MR 1174740, DOI 10.1007/978-94-011-3604-4
Jakob Levitzki, On normal products of algebras, Ann. of Math. (2) 33 (1932), no. 3, 377–402. MR 1503062, DOI 10.2307/1968521
Martin Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), no. 2, 89–132. MR 571248, DOI 10.1007/BF02760553
Martin Lorenz and D. S. Passman, Prime ideals in group algebras of polycyclic-by-finite groups, Proc. London Math. Soc. (3) 43 (1981), no. 3, 520–543. MR 635568, DOI 10.1112/plms/s3-43.3.520
Yôichi Miyashita, On Galois extensions and crossed products, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970), 97–121. MR 0271163
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Susan Montgomery, Outer automorphisms of semi-prime rings, J. London Math. Soc. (2) 18 (1978), no. 2, 209–220. MR 509936, DOI 10.1112/jlms/s2-18.2.209
Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
Susan Montgomery, Prime ideals and group actions in noncommutative algebras, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 103–124. MR 999986, DOI 10.1090/conm/088/999986
John A. Moody, Induction theorems for infinite groups, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 113–116. MR 888884, DOI 10.1090/S0273-0979-1987-15527-3
Yoshiomi Nakagami and Masamichi Takesaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Mathematics, vol. 731, Springer, Berlin, 1979. MR 546058
Tadasi Nakayama and Gorô Azumaya, On irreducible rings, Ann. of Math. (2) 48 (1947), 949–965. MR 24891, DOI 10.2307/1969388
Emmy Noether, Nichtkommutative Algebra, Math. Z. 37 (1933), no. 1, 514–541 (German). MR 1545411, DOI 10.1007/BF01474591
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
D. S. Passman, Semiprime crossed products, Houston J. Math. 11 (1985), no. 2, 257–267. MR 792201
D. S. Passman, Prime ideals in polycyclic crossed products, Trans. Amer. Math. Soc. 301 (1987), no. 2, 737–759. MR 882713, DOI 10.1090/S0002-9947-1987-0882713-9
Declan Quinn, Integrality over fixed rings, J. London Math. Soc. (2) 40 (1989), no. 2, 206–214. MR 1044269, DOI 10.1112/jlms/s2-40.2.206
Marc A. Rieffel, Actions of finite groups on $C^{\ast }$-algebras, Math. Scand. 47 (1980), no. 1, 157–176. MR 600086, DOI 10.7146/math.scand.a-11882
J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), no. 3, 385–447. MR 491797, DOI 10.1112/plms/s3-36.3.385
B. L. van der Waerden, A history of algebra, Springer-Verlag, Berlin, 1985. From al-Khwārizmī to Emmy Noether. MR 803326, DOI 10.1007/978-3-642-51599-6
J. H. M. Wedderburn, A type of primitive algebra, Trans. Amer. Math. Soc. 15 (1914), no. 2, 162–166. MR 1500970, DOI 10.1090/S0002-9947-1914-1500970-5
[Z] A. E. Zaleskiĭ, Irreducible representations of finitely-generated nilpotent torsion-free groups, Math. Notes 9 (1971), 117-123.
A. E. Zalesskiĭ and O. M. Neroslavskiĭ, Simple Noetherian rings, Vescī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 5 (1975), 38–42, 138 (Russian). MR 0389968
- [AH] A. A. Albert and H. Hasse, A determination of all normal division algebras over an algebraic number field, Trans. Amer. Math. Soc. 34 (1932), 722-726. MR 1501659
- [Am] S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408-420. MR 318216
- [Az] G. Azumaya, New foundation of the theory of simple rings, Proc. Japan Acad. 22 (1946), 325-332. MR 42387
- [B] A. A. Bovdi, Crossed products of a semigroup and a ring, Soviet Math. Dokl. 2 (1961), 438-440. MR 132766
- [BHN] R. Brauer, H. Hasse, and E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, J. für Math. 167 (1932), 399-404.
- [CM] M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), 237-258. MR 728711
- [C] P. M. Cohn, Free rings and their relations, 2nd ed., Academic Press, London, 1985. MR 800091
- [D1] L. E. Dickson, Abstract, Bull. Amer. Math. Soc. (N.S.) 12 (1906), 441-442.
- [D2] L. E. Dickson, Linear associative algebras and abelian equations, Trans. Amer. Math. Soc. 15 (1914), 31-46. MR 1500963
- [E] G. A. Elliot, Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal. 23 (1976), 1-10. MR 440372
- [FM] J. W. Fisher and S. Montgomery, Semiprime skew group rings, J. Algebra 52 (1978), 241-247. MR 480616
- [FO] J. W. Fisher and J. Osterburg, Some results on rings with finite group actions, Proceedings of 1976 Ohio Conference, Lecture Notes in Math., vol. 25, Marcel Dekker, New York, 1976. MR 442022
- [J1] N. Jacobson, The fundamental theorem of the Galois theory for quasi-fields, Ann. of Math. (2)41 (1940), 1-7. MR 1219
- [J2] N. Jacobson, Theory of rings, Math. Surveys, vol. 2, Amer. Math. Soc. Providence, RI, 1943. MR 8601
- [Kh1] V. K. Kharchenko, Galois theory of semiprime rings, Algebra i Logika 16 (1977), 313-363 (English transl. 1978, 208-258). MR 573067
- [Kh2] V. K. Kharchenko, Algebras of invariants of free algebras, Algebra i Logika 17 (1978), 478-487 (English transl. 1979, 316-321). MR 538309
- [Kh3] V. K. Kharchenko, Automorphisms and derivations of associative rings, Reidel (to appear). MR 1174740
- [L] J. Levitzki, On normal products of algebras, Ann. of Math. (2) 33 (1932), 377-402. MR 1503062
- [LP1] M. Lorenz and D. S. Passman, Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), 89-132 (addendum 35 (1980), 311-322). MR 571248
- [LP2] M. Lorenz and D. S. Passman, Prime ideals in group algebras of polycyclic-by-finite groups, Proc. London Math. Soc. 43 (1981), 520-543. MR 635568
- [Mi] Y. Miyashita, On Galois extensions and crossed products, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970), 97-121. MR 271163
- [McR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley-Intersceience, New York, 1987. MR 934572
- [M1] S. Montgomery, Outer automorphisms of semiprime rings, J. London Math. Soc. 18 (1978), 209-220. MR 509936
- [M2] S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math., vol. 818, Springer-Verlag, Berlin and New York, 1980. MR 590245
- [M3] S. Montgomery, Prime ideals and group actions in noncommutative algebras, Invariant Theory, Contemp. Math., vol. 88, Amer. Math. Soc. Providence, RI, (1989), pp. 103-124. MR 999986
- [Mo] J. A. Moody, Induction theorems for infinite groups, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 113-116. MR 888884
- [NT] Y. Nakagami and M. Takasaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Math., vol. 731, Springer-Verlag, Berlin and New York, 1979. MR 546058
- [NA] T. Nakayama and G. Azumaya, On irreducible rings, Ann. of Math. (2)48 (1947), 949-965. MR 24891
- [N] E. Noether, Nichtkommutative algebra, Math. Z. 37 (1933), 514-541. MR 1545411
- [P1] D. S. Passman, The algebraic structure of group rings, Wiley-Interscience, New York, 1977. MR 470211
- [P2] D. S. Passman, Semiprime crossed products, Houston J. Math. 11(1985), 257-268. MR 792201
- [P3] D. S. Passman, Prime ideals in polycyclic crossed products, Trans. Amer. Math. Soc. 301 (1987), 737-759. MR 882713
- [Q] D. Quinn, Integrality over fixed rings, J. London Math. Soc. 40 (1989), 206-214. MR 1044269
- [Ri] M. A. Rieffel, Actions of finite groups on C*-algebras, Math. Scand. 47 (1980), 157-176. MR 600086
- [R] J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. 36 (1978), 385-447. MR 491797
- [vdW] B. L. van der Waerden, A history of algebra, Springer-Verlag, Berlin and New York, 1985. MR 803326
- [W] J. H. M. Wedderburn, A type of primitive algebra, Trans. Amer. Math. Soc. 15 (1914), 162-166. MR 1500970
- [Z] A. E. Zaleskiĭ, Irreducible representations of finitely-generated nilpotent torsion-free groups, Math. Notes 9 (1971), 117-123.
- [ZN] A. E. Zaleskiĭand O. M. Neroslavskii, On simple Noetherian rings, Proc. Akad. Bylorussia SSR (1975), 38-42. (Russian) MR 389968
Review Information:
Reviewer:
Susan Montgomery
Journal:
Bull. Amer. Math. Soc.
24 (1991), 391-402
DOI:
https://doi.org/10.1090/S0273-0979-1991-16044-1