Krull dimension of modules over involution rings. II
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- by K. I. Beidar, E. R. Puczylowski and P. F. Smith
- Proc. Amer. Math. Soc. 125 (1997), 355-361
- DOI: https://doi.org/10.1090/S0002-9939-97-03724-6
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Abstract:
Let $R$ be a ring with involution and invertible 2, and let $\bar S$ be the subring of $R$ generated by the symmetric elements in $R$. The following questions of Lanski are answered positively:
[(i)] Must $\bar S$ have Krull dimension when $R$ does?
[(ii)] Is every Artinian $R$-module Artinian as an $\bar S$-module?
References
- S. A. Amitsur, Rings with involution, Israel J. Math. 6 (1968), 99–106. MR 238896, DOI 10.1007/BF02760175
- K. I. Beĭdar, E. R. Puczyłowski, and P. F. Smith, Krull dimension of modules over involution rings, Proc. Amer. Math. Soc. 121 (1994), no. 2, 391–397. MR 1184081, DOI 10.1090/S0002-9939-1994-1184081-3
- K. I. Beĭdar and V. T. Markov, A semiprime PI ring which has a faithful module with Krull dimension is a Goldie ring, Uspekhi Mat. Nauk 48 (1993), no. 6(294), 141–142 (Russian); English transl., Russian Math. Surveys 48 (1993), no. 6, 158. MR 1264158, DOI 10.1070/RM1993v048n06ABEH001096
- C. L. Chuang and Pjek Hwee Lee, Noetherian rings with involution, Chinese J. Math. 5 (1977), no. 1, 15–19. MR 453800
- Robert Gordon and J. C. Robson, Krull dimension, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR 0352177
- I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
- Charles Lanski, On the relationship of a ring and the subring generated by its symmetric elements, Pacific J. Math. 44 (1973), 581–592. MR 321966
- Charles Lanski, Chain conditions in rings with involution, J. London Math. Soc. (2) 9 (1974/75), 93–102. MR 360676, DOI 10.1112/jlms/s2-9.1.93
- Charles Lanski, Chain conditions in rings with involution. II, J. London Math. Soc. (2) 18 (1978), no. 3, 421–428. MR 518226, DOI 10.1112/jlms/s2-18.3.421
- Charles Lanski, Gabriel dimension and rings with involution, Houston J. Math. 4 (1978), no. 3, 397–415. MR 514253
- Pjek Hwee Lee, On subrings of rings with involution, Pacific J. Math. 60 (1975), no. 2, 131–147. MR 396657
- V.T. Markov, On PI rings having a faithful module with Krull dimension (to appear).
- Susan Montgomery, A structure theorem and a positive-definiteness condition in rings with involution, J. Algebra 43 (1976), no. 1, 181–192. MR 424863, DOI 10.1016/0021-8693(76)90152-6
- Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
- Louis Halle Rowen, On rings with central polynomials, J. Algebra 31 (1974), 393–426. MR 349744, DOI 10.1016/0021-8693(74)90122-7
Bibliographic Information
- K. I. Beidar
- Affiliation: Department of Mathematics, Moscow State University, Moscow, Russia
- Address at time of publication: National Cheng–Kung University, Department of Mathematics, Tainan, Taiwan
- Email: t14270@sparc1.cc.ncku.edu.tw
- E. R. Puczylowski
- Affiliation: Institute of Mathematics, University of Warsaw, Warsaw, Poland
- Email: edmundp@mimuw.edu.pl
- P. F. Smith
- Affiliation: Department of Mathematics, University of Glasgow, Glasgow, Scotland
- Email: pfs@maths.gla.ac.uk
- Received by editor(s): August 23, 1995
- Additional Notes: The research of the second author was partially supported by KBN grant 2 P301 035 06.
- Communicated by: Ken Goodearl
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 355-361
- MSC (1991): Primary 16W10, 16P60
- DOI: https://doi.org/10.1090/S0002-9939-97-03724-6
- MathSciNet review: 1371115