New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings
HTML articles powered by AMS MathViewer
- by Jenő Szigeti PDF
- Proc. Amer. Math. Soc. 125 (1997), 2245-2254 Request permission
Abstract:
We construct the so-called right adjoint sequence of an $n\times n$ matrix over an arbitrary ring. For an integer $m\geq 1$ the right $m$-adjoint and the right $m$-determinant of a matrix is defined by the use of this sequence. Over $m$-Lie nilpotent rings a considerable part of the classical determinant theory, including the Cayley-Hamilton theorem, can be reformulated for our right adjoints and determinants. The new theory is then applied to derive the PI of algebraicity for matrices over the Grassmann algebra.References
- A. Berele and A. Regev, Applications of hook Young diagrams to P.I. algebras, J. Algebra 82 (1983), no. 2, 559–567. MR 704771, DOI 10.1016/0021-8693(83)90167-9
- Allan Berele, Magnum P.I, Israel J. Math. 51 (1985), no. 1-2, 13–19. MR 804472, DOI 10.1007/BF02772954
- Allan Berele, Azumaya-like properties of verbally prime algebras, J. Algebra 133 (1990), no. 2, 272–276. MR 1067407, DOI 10.1016/0021-8693(90)90270-X
- A. A. Bovdi and I. I. Khripta, Generalized Lie nilpotent group rings, Mat. Sb. (N.S.) 129(171) (1986), no. 1, 154–158, 160 (Russian); English transl., Math. USSR-Sb. 57 (1987), no. 1, 165–169. MR 830101, DOI 10.1070/SM1987v057n01ABEH003061
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- Issai Kantor and Ivan Trishin, On a concept of determinant in the supercase, Comm. Algebra 22 (1994), no. 10, 3679–3739. MR 1280094, DOI 10.1080/00927879408825050
- A.R. Kemer, Varieties of $\mathbf {Z}_2$-graded algebras, Math. USSR Izv. 25 (1985), 359-374.
- A. R. Kemer, The standard identity in characteristic $p$: a conjecture of I. B. Volichenko, Israel J. Math. 81 (1993), no. 3, 343–355. MR 1231198, DOI 10.1007/BF02764837
- Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR 1108620, DOI 10.1090/mmono/087
- D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. MR 325658, DOI 10.1090/S0002-9947-1973-0325658-5
- I. B. S. Passi, D. S. Passman, and S. K. Sehgal, Lie solvable group rings, Canadian J. Math. 25 (1973), 748–757. MR 325746, DOI 10.4153/CJM-1973-076-4
- Angel P. Popov, On the identities of the matrices over the Grassmann algebra, J. Algebra 168 (1994), no. 3, 828–852. MR 1293628, DOI 10.1006/jabr.1994.1257
Additional Information
- Jenő Szigeti
- Affiliation: Institute of Mathematics, University of Miskolc, Miskolc-Egyetemváros, 3515 Hungary
- MR Author ID: 169785
- Email: matszj@gold.uni-miskolc.hu
- Received by editor(s): December 19, 1995
- Received by editor(s) in revised form: March 6, 1996
- Additional Notes: Supported by OTKA of Hungary, grant no. T7558, and by the Computer and Automation Institute of the Hungarian Academy of Science.
- Communicated by: Lance W. Small
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2245-2254
- MSC (1991): Primary 16A38, 15A15; Secondary 15A33
- DOI: https://doi.org/10.1090/S0002-9939-97-03868-9
- MathSciNet review: 1389540