On invertible bimodules and automorphisms of noncommutative rings

Guralnick, Robert M.; Montgomery, Susan

doi:10.1090/S0002-9947-1994-1150014-3

On invertible bimodules and automorphisms of noncommutative rings
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by Robert M. Guralnick and Susan Montgomery

Trans. Amer. Math. Soc. 341 (1994), 917-937

DOI: https://doi.org/10.1090/S0002-9947-1994-1150014-3

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Abstract:

In this article, we attempt to generalize the result that for a commutative ring $R$ the outer automorphism group of $R$-automorphisms of ${M_n}(R)$ is abelian of exponent $n$. It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.

References

M. Artin and W. Schelter, Integral ring homomorphisms, Adv. in Math. 39 (1981), no. 3, 289–329. MR 614165, DOI 10.1016/0001-8708(81)90005-0
Hyman Bass and Robert Guralnick, Projective modules with free multiples and powers, Proc. Amer. Math. Soc. 96 (1986), no. 2, 207–208. MR 818444, DOI 10.1090/S0002-9939-1986-0818444-5
Amiram Braun, An additivity principle for p.i. rings, J. Algebra 96 (1985), no. 2, 433–441. MR 810539, DOI 10.1016/0021-8693(85)90020-1

Some notes on a theorem of Farkas

Kenneth A. Brown, The representation theory of Noetherian rings, Noncommutative rings (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 24, Springer, New York, 1992, pp. 1–25. MR 1230215, DOI 10.1007/978-1-4613-9736-6_{1}
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
Daniel R. Farkas, Groups acting on affine algebras, Trans. Amer. Math. Soc. 310 (1988), no. 2, 485–497. MR 940913, DOI 10.1090/S0002-9947-1988-0940913-4
Robert M. Guralnick, The genus of a module. II. Roĭter’s theorem, power cancellation and extension of scalars, J. Number Theory 26 (1987), no. 2, 149–165. MR 889381, DOI 10.1016/0022-314X(87)90075-8
Robert M. Guralnick, A question of Stafford about affine pi algebras, Comm. Algebra 18 (1990), no. 9, 3055–3057. MR 1063350, DOI 10.1080/00927879008824060
Robert M. Guralnick, Bimodules over PI rings, Methods in module theory (Colorado Springs, CO, 1991) Lecture Notes in Pure and Appl. Math., vol. 140, Dekker, New York, 1993, pp. 117–134. MR 1203803
William H. Gustafson and Klaus W. Roggenkamp, Automorphisms and Picard groups for hereditary orders, Visiting scholars’ lectures—1987 (Lubbock, TX), Texas Tech Univ. Math. Ser., vol. 15, Texas Tech Univ., Lubbock, TX, 1988, pp. 37–51. MR 983895
William H. Gustafson and Klaus W. Roggenkamp, A Mayer-Vietoris sequence for Picard groups, with applications to integral group rings of dihedral and quaternion groups, Illinois J. Math. 32 (1988), no. 3, 375–406. MR 947034
Bao-Ping Jia, Splitting of rank-one valuations, Comm. Algebra 19 (1991), no. 3, 777–794. MR 1102985, DOI 10.1080/00927879108824169
Bao-Ping Jia, Splitting of primes in extension domains, Comm. Algebra 19 (1991), no. 9, 2603–2623. MR 1125193, DOI 10.1080/00927879108824283
Irving Kaplansky, Commutative rings, Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945

Théorie de la descente et Algèbres d’Azumaya, Lecture Notes in Math., vol. 389, Springer-Verlag, Berlin and New York, 1974

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, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
S. Montgomery, A generalized Picard group for prime rings, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 55–63. MR 1171225
S. Montgomery and D. S. Passman, Outer Galois theory of prime rings, Rocky Mountain J. Math. 14 (1984), no. 2, 305–318. MR 747279, DOI 10.1216/RMJ-1984-14-2-305
Alex Rosenberg and Daniel Zelinsky, Automorphisms of separable algebras, Pacific J. Math. 11 (1961), 1109–1117. MR 148709
S. P. Smith, Can the Weyl algebra be a fixed ring?, Proc. Amer. Math. Soc. 107 (1989), no. 3, 587–589. MR 962247, DOI 10.1090/S0002-9939-1989-0962247-0
S. P. Smith, An example of a ring Morita equivalent to the Weyl algebra $A_{1}$, J. Algebra 73 (1981), no. 2, 552–555. MR 640048, DOI 10.1016/0021-8693(81)90334-3
J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra, Trans. Amer. Math. Soc. 299 (1987), no. 2, 623–639. MR 869225, DOI 10.1090/S0002-9947-1987-0869225-3
Roger Wiegand, Nilpotent elements in Grothendieck rings, Illinois J. Math. 32 (1988), no. 2, 246–262. MR 945862