Finiteness properties of differential polynomials
Resource
Linear Algebra and its Applications 430 (8-9): 2030-2041
Journal
Linear Algebra and Its Applications
Journal Volume
430
Journal Issue
8-9
Pages
2030-2041
Date Issued
2009
Author(s)
Abstract
Let R be a prime ring with extended centroid C and let φ{symbol} (XiΔj) be a reduced differential polynomial with coefficients in Q, the symmetric Martindale quotient ring of R, and with zero constant term. Let Aφ{symbol} = {φ{symbol} (xiΔj) | xi ∈ R} and Bφ{symbol} = {φ{symbol} (xij) | xij ∈ R}. We prove that the finiteness of Aφ{symbol} and the finite-dimensionality of the C-span of Aφ{symbol} are equivalent to that of Bφ{symbol} and that of the C-span of Bφ{symbol}, respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials. Let δ and d be two derivations of R, L a Lie ideal of R and ρ a right ideal of R. As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of d (ρ), d (L) and δ d (L) and for the finite-dimensionality of the C-spans of d (ρ), d (L) and δ d (L). © 2008 Elsevier Inc. All rights reserved.
Type
journal article
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