Trace-like functions on rings with no nilpotent elements

Cohen, M.; Montgomery, Susan

doi:10.1090/S0002-9947-1982-0664033-6

Trace-like functions on rings with no nilpotent elements
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by M. Cohen and Susan Montgomery PDF

Trans. Amer. Math. Soc. 273 (1982), 131-145 Request permission

Abstract:

Let $R$ be a ring with no nilpotent elements, with extended center $C$, and let $E$ be the set of idempotents in $C$. Our first main result is that for any finite group $G$ acting as automorphisms of $R$, there exist a finite set $L \subseteq E$ and an ${R^G}$-bimodule homomorphism $\tau :R \to {(RL)^G}$ such that $\tau (R)$ is an essential ideal of ${(RE)^G}$. This theorem is applied to show the following: if $R$ is a Noetherian, affine $PI$-algebra (with no nilpotent elements) over the commutative Noetherian ring $A$, and $G$ is a finite group of $A$-automorphisms of $R$ such that ${R^G}$ is Noetherian, then ${R^G}$ is affine over $A$.

References

S. A. Amitsur, On rings of quotients, Symposia Mathematica, Vol. VIII (Convegno sulleAlgebre Associative, INDAM, Rome, 1970) Academic Press, London, 1972, pp. 149–164. MR 0332855
S. A. Amitsur and Lance W. Small, Prime $PI$-rings, Bull. Amer. Math. Soc. 83 (1977), no. 2, 249–251. MR 435134, DOI 10.1090/S0002-9904-1977-14285-7
G. M. Bergman and I. M. Isaacs, Rings with fixed-point-free group actions, Proc. London Math. Soc. (3) 27 (1973), 69–87. MR 346017, DOI 10.1112/plms/s3-27.1.69
C. L. Chuang and Pjek Hwee Lee, Noetherian rings with involution, Chinese J. Math. 5 (1977), no. 1, 15–19. MR 453800
Miriam Cohen, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98 (1982), no. 1, 37–54. MR 644936
M. Cohen and S. Montgomery, The normal closure of a semiprime ring, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978) Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, 1979, pp. 43–59. MR 563284
M. Cohen and Susan Montgomery, Trace functions for finite automorphism groups of rings, Arch. Math. (Basel) 35 (1980), no. 6, 516–527 (1981). MR 604250, DOI 10.1007/BF01235376
Daniel R. Farkas and Robert L. Snider, Noetherian fixed rings, Pacific J. Math. 69 (1977), no. 2, 347–353. MR 444714
R. M. Guralnick, I. M. Isaacs, and D. S. Passman, Nonexistence of partial traces for group actions, Rocky Mountain J. Math. 11 (1981), no. 3, 395–405. MR 722573, DOI 10.1216/RMJ-1981-11-3-395

Galois extensions and quotient rings

13

V. K. Harčenko, Generalized identities with automorphisms, Algebra i Logika 14 (1975), no. 2, 215–237, 241 (Russian). MR 0399153
V. K. Harčenko, Fixed elements of a finite group action on a semiprime ring, Algebra i Logika 14 (1975), no. 3, 328–344, 369–370 (Russian). MR 0429998
V. K. Harčenko, The Galois theory of semiprime rings, Algebra i Logika 16 (1977), no. 3, 313–363 (Russian). MR 0573067
Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. MR 238897, DOI 10.1016/0021-8693(69)90029-5
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, Automorphism groups of rings with no nilpotent elements, J. Algebra 60 (1979), no. 1, 238–248. MR 549109, DOI 10.1016/0021-8693(79)90119-4
S. Montgomery and L. W. Small, Fixed rings of Noetherian rings, Bull. London Math. Soc. 13 (1981), no. 1, 33–38. MR 599637, DOI 10.1112/blms/13.1.33
David Mumford, Hilbert’s fourteenth problem–the finite generation of subrings such as rings of invariants, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc.., Providence, R.I., 1976, pp. 431–444. MR 0435076
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