Derivations and right ideals of algebras
Journal
Linear Algebra and Its Applications
Journal Volume
432
Journal Issue
11
Pages
2773-2781
Date Issued
2010
Author(s)
Abstract
Let R be a K-algebra acting densely on VD, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f (X1, ..., Xt) be a nonzero polynomial over K with constant term 0 such that μ R ≠ 0 for some coefficient μ of f (X1, ..., Xt). Suppose that d : R → R is a nonzero derivation. It is proved that if rank d (f (x1, ..., xt)) ≤ m for all x1, ..., xt ∈ ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d = ad (b) for some b ∈ End (VD) of finite rank. In addition, if f (X1, ..., Xt) is multilinear, then b can be chosen such that rank (b) ≤ 2 (6 t + 13) m + 2. © 2009 Elsevier Inc. All rights reserved.
Type
journal article