New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings

Author:
Jeno Szigeti

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct the so-called right adjoint sequence of an matrix over an arbitrary ring. For an integer the right -adjoint and the right -determinant of a matrix is defined by the use of this sequence. Over -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.

**1.**A. Berele and A. Regev,*Applications of hook Young diagrams to PI-algebras,*J. Algebra**82**(1983), 559-567. MR**84g:16012****2.**A. Berele,*Magnum P.I.,*Israel J. Math.**51**(1985), 13-19. MR**87b:16019****3.**A. Berele,*Azumaya-like properties of verbally prime algebras*, J. Algebra**133**No.2, (1990), 272-276. MR**91g:16014****4.**B. Bódi and I.I. Khripta,*Generalized Lie nilpotent group rings,*Mat. Sbornik**129**No.1, (1986), 154-158. MR**87e:16031****5.**N. Jacobson,*Structure theory of algebraic algebras of bounded degree*, Ann. of Math.**46**(1945), 695-707. MR**7:238c****6.**I. Kantor and I. Trishin,*On a concept of determinant in the supercase*, Comm. Algebra**22**(10), (1994), 3679-3739. MR**95c:15062****7.**A.R. Kemer,*Varieties of -graded algebras*, Math. USSR Izv.**25**(1985), 359-374.**8.**A.R. Kemer,*The standard identity in characteristic p: a conjecture of I. B. Volichenko*, Israel J. Math.**8**(1993), 343-355. MR**94f:16040****9.**A.R. Kemer,*Ideals of Identities of Associative Algebras*, Translations of Math. Monographs, Vol.87 (1991), AMS Providence, Rhode Island. MR**92f:16031****10.**D. Krakowsky and A. Regev,*The polynomial identities of the Grassmann algebra*, Trans. Amer. Math. Soc.**181**(1973), 429-438. MR**48:4005****11.**I.B.S. Passi, D.S. Passman and S.K. Sehgal,*Lie solvable group rings*, Can. J. Math.**25**No. 4, (1973), 748-757. MR**48:4092****12.**A. Popov,*On the identities of the matrices over the Grassmann algebra*, J. Algebra**168**No. 3, (1994), 828-852. MR**95h:16029**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
16A38,
15A15,
15A33

Retrieve articles in all journals with MSC (1991): 16A38, 15A15, 15A33

Additional Information

**Jeno Szigeti**

Affiliation:
Institute of Mathematics, University of Miskolc, Miskolc-Egyetemváros, 3515 Hungary

Email:
matszj@gold.uni-miskolc.hu

DOI:
https://doi.org/10.1090/S0002-9939-97-03868-9

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

Article copyright:
© Copyright 1997
American Mathematical Society