# Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X 1,...,X t ) a multilinear polynomial not central-valued on R. Suppose d(f(x 1,...,x t )) is either invertible or nilpotent for all x 1,...,x t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

# Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

, Volume 7 (1) – Jan 1, 2000
6 pages      /lp/springer-journals/derivations-with-invertible-or-nilpotent-values-on-a-multilinear-nHRubtkWQW
Publisher
Springer Journals
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s10011-000-0093-2
Publisher site
See Article on Publisher Site

### Abstract

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X 1,...,X t ) a multilinear polynomial not central-valued on R. Suppose d(f(x 1,...,x t )) is either invertible or nilpotent for all x 1,...,x t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.

### Journal

Algebra ColloquiumSpringer Journals

Published: Jan 1, 2000