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On homomorphisms and cv-polynomials between iterated Ore extensions

María Camila Ramírez, Armando Reyes

Abstract

Motivated by the study of homomorphisms and cv-polynomials presented by Rimmer \cite{Rimmer1978} in the case of Ore extensions of automorphism type, Ferrero and Kishimoto \cite{FerreroKishimoto1980} and Kikumasa \cite{Kikumasa1990} in the setting of these extensions of derivation type, and Lam and Leroy \cite{LamLeroy1992} in the context of Ore extensions of mixed type over division rings, in this paper we investigate these both notions for iterations of these extensions. We show that the iteration of some of the results presented in those papers is non-trivial, and illustrate our treatment with several noncommutative algebras.

On homomorphisms and cv-polynomials between iterated Ore extensions

Abstract

Motivated by the study of homomorphisms and cv-polynomials presented by Rimmer \cite{Rimmer1978} in the case of Ore extensions of automorphism type, Ferrero and Kishimoto \cite{FerreroKishimoto1980} and Kikumasa \cite{Kikumasa1990} in the setting of these extensions of derivation type, and Lam and Leroy \cite{LamLeroy1992} in the context of Ore extensions of mixed type over division rings, in this paper we investigate these both notions for iterations of these extensions. We show that the iteration of some of the results presented in those papers is non-trivial, and illustrate our treatment with several noncommutative algebras.
Paper Structure (6 sections, 17 theorems, 62 equations)

This paper contains 6 sections, 17 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two-step iterated Ore extensions
  4. Three-step iterated Ore extensions
  5. $n$-step iterated Ore extensions
  6. Future work

Key Result

Proposition 3.6

Let $p(x) = \sum_{i = 0}^{n} r_ix^{i} \in D[x;\sigma, \delta]$ be of degree $n\ge 0$, and let $(\sigma', \delta')$ be any quasi-derivation on $D$.

Theorems & Definitions (52)

  • Example 2.1
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Example 3.4
  • Example 3.5
  • Proposition 3.6: LamLeroy1992
  • Theorem 3.8
  • proof
  • Remark 3.9
  • ...and 42 more