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Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials.

V De Filippis1, G Scudo1, L Sorrenti1

  • 1Department of Mathematics, Faculty of Sciences, University of Messina, Via F. Stagno D'Alcontres 31, 98166 Messina, Italy.

International Scholarly Research Notices
|July 6, 2016
PubMed
Summary

This study investigates generalized derivations on prime rings. If a specific polynomial equation holds, it implies the derivation is either trivial, of a specific form, or the polynomial is centrally valued.

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Area of Science:

  • Abstract Algebra
  • Ring Theory
  • Noncommutative Algebra

Background:

  • Generalized derivations are crucial in understanding the algebraic structure of rings.
  • Multilinear polynomials play a significant role in exploring properties of derivations and generalized derivations.
  • Prime rings with characteristic not equal to 2 provide a fundamental framework for these investigations.

Purpose of the Study:

  • To analyze the conditions under which a generalized derivation satisfies a specific polynomial equation involving its iterated application.
  • To determine the possible structures of the generalized derivation and the properties of the associated multilinear polynomial.
  • To extend existing results in ring theory concerning derivations and generalized derivations.

Main Methods:

  • Utilized concepts from ring theory, including prime rings, extended centroids, and Utumi quotient rings.
  • Applied techniques related to multilinear polynomials and their behavior over the extended centroid.
  • Investigated the algebraic implications of the given functional equation a[F(f(r1,…,rn)), f(r1,…,rn)]b = 0.

Main Results:

  • Established that the given condition leads to one of five possibilities for the generalized derivation F and the polynomial f.
  • These possibilities include trivial cases (a=0 or b=0), F being a scalar multiple of the identity, or F having a specific form involving an element from the Utumi quotient ring.
  • The results also highlight conditions on the multilinear polynomial, such as being centrally valued or having specific properties related to the element q.

Conclusions:

  • The study provides a comprehensive classification of generalized derivations satisfying the given polynomial identity.
  • The findings contribute to a deeper understanding of the interplay between generalized derivations, multilinear polynomials, and the structure of prime rings.
  • This research offers a foundation for further exploration into algebraic structures and functional identities in ring theory.