Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T17:04:24.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneous Polynomials, Centralizers and Derivations in Rings

Published online by Cambridge University Press:  20 November 2018

Onofrio Mario Divincenzo  and
Rosa Sagona
Onofrio Mario Divincenzo
Affiliation:
Dipartimento di Matematica, Universita della Basilicata, via N. Sauro 85, 85100 Potenza, Italy
Rosa Sagona
Affiliation:
Dipartimento di Matematica, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let d be a non-zero derivation on a primitive ring R and ƒ(x1,…, xn) a homogeneous polynomial of degree m. We prove that the condition d(ƒ(r1,…, rn)t) = 0, for all r1,…, rn ∈ R, with t depending on r1,…, rn, forces R to be a finite dimensional central simple algebra and ƒ power-central valued on R. We also obtain bounds on [R : Z(R)] in terms of m.

Keywords

16A3816A72
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 1 , 01 February 1993 , pp. 22 - 32
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bergen, J. and Giambruno, A., -radical extensions of rings, Rend. Sem. Mat. Univ. Padova 77(1987), 125–133.Google Scholar
2. Di, O.M. Vincenzo, Derivations and multilinear polynomials, Rend. Sem. Mat. Univ. Padova 81(1989), 209–219.Google Scholar
3. Felzenszwalb, B. and Giambruno, A., Centralizers and multilinear polynomials in non-commutative rings, J. London Math. Soc. (2) 19(1979), 417–428.Google Scholar
4. Felzenszwalb, B., Periodic and nil polynomials in rings, Canad. Math. Bull. (4) 23(1980), 473–476.Google Scholar
5. Giambruno, A., Rings f-radical over subrings P.I., Rend. Mat. Roma (1) (VI) 13(1980), 105–113.Google Scholar
6. Herstein, I.N., Noncommutative rings, Cams Mathematical Monographs, Math. Assoc. Amer. 1968.Google Scholar
7. Herstein, I.N., Rings with involution, Univ. Chicago Press, Chicago, 1976.Google Scholar
8. Herstein, I.N., A Theorem on invariant subrings, J. Algebra 83(1983), 26–32.Google Scholar
9. Herstein, I.N., Procesi, C. and Shacher, M., Algebraic valued functions on noncommutative rings, J. Algebra 36(1975), 128–150.Google Scholar
10. Rowen, L.H., Polynomial identities in ring theory, Academic Press, New York, 1973.Google Scholar