Actions of linearly reductive groups on PI-algebras

Vonessen, Nikolaus

doi:10.1090/S0002-9947-1993-1076618-3

Actions of linearly reductive groups on PI-algebras
HTML articles powered by AMS MathViewer

by Nikolaus Vonessen PDF

Trans. Amer. Math. Soc. 335 (1993), 425-442 Request permission

Abstract:

Let $G$ be a linearly reductive group acting rationally on a ${\text {PI}}$-algebra $R$. We study the relationship between $R$ and the fixed ring ${R^G}$ , generalizing earlier results obtained under the additional hypothesis that $R$ is affine.

References

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
George M. Bergman and Lance W. Small, P.I. degrees and prime ideals, J. Algebra 33 (1975), 435–462. MR 360682, DOI 10.1016/0021-8693(75)90112-X
Walter Borho, On the Joseph-Small additivity principle for Goldie ranks, Compositio Math. 47 (1982), no. 1, 3–29. MR 668778
Amiram Braun and Nikolaus Vonessen, Integrality for PI-rings, J. Algebra 151 (1992), no. 1, 39–79. MR 1182014, DOI 10.1016/0021-8693(92)90131-5
William Chin, Actions of solvable algebraic groups on noncommutative rings, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 29–38. MR 1144026, DOI 10.1090/conm/124/1144026
Daniel R. Farkas and Robert L. Snider, Noetherian fixed rings, Pacific J. Math. 69 (1977), no. 2, 347–353. MR 444714
A. Joseph and L. W. Small, An additivity principle for Goldie rank, Israel J. Math. 31 (1978), no. 2, 105–114. MR 516246, DOI 10.1007/BF02760541
Edward S. Letzter, Prime ideals in finite extensions of Noetherian rings, J. Algebra 135 (1990), no. 2, 412–439. MR 1080857, DOI 10.1016/0021-8693(90)90299-4
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
Susan Montgomery, Prime ideals in fixed rings, Comm. Algebra 9 (1981), no. 4, 423–449. MR 605031, DOI 10.1080/00927878108822591

Prime ideals and group actions in non-commutative algebras

88

S. Montgomery and L. W. Small, Integrality and prime ideals in fixed rings of P.I. rings, J. Pure Appl. Algebra 31 (1984), no. 1-3, 185–190. MR 738213, DOI 10.1016/0022-4049(84)90084-7
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
William Schelter, Integral extensions of rings satisfying a polynomial identity, J. Algebra 40 (1976), no. 1, 245–257. MR 417238, DOI 10.1016/0021-8693(76)90095-8
Nikolaus Vonessen, Actions of linearly reductive groups on affine PI algebras, Mem. Amer. Math. Soc. 81 (1989), no. 414, iv+106. MR 990363, DOI 10.1090/memo/0414
N. Vonessen, Prime ideals in intermediate integral centralizing extensions of PI-rings, Comm. Algebra 19 (1991), no. 3, 795–802. MR 1102986, DOI 10.1080/00927879108824170
R. B. Warfield Jr., Prime ideals in ring extensions, J. London Math. Soc. (2) 28 (1983), no. 3, 453–460. MR 724714, DOI 10.1112/jlms/s2-28.3.453