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Exploring Functional Identities: From Division Rings to Matrix Algebras

Daniel Kawai, Bruno Leonardo Macedo Ferreira

Abstract

In this paper, we tackle unresolved inquiries by Ferreira et al. \cite{bruno} in their recent publication, ``Functional Identity on Division Algebras". We delve into the intricate behavior of additive functions on matrix algebras over division rings through rigorous analysis and theorem-proving. Our findings offer valuable insights into the nature of these functions and their implications for algebraic structures.

Exploring Functional Identities: From Division Rings to Matrix Algebras

Abstract

In this paper, we tackle unresolved inquiries by Ferreira et al. \cite{bruno} in their recent publication, ``Functional Identity on Division Algebras". We delve into the intricate behavior of additive functions on matrix algebras over division rings through rigorous analysis and theorem-proving. Our findings offer valuable insights into the nature of these functions and their implications for algebraic structures.
Paper Structure (2 sections, 4 theorems, 72 equations)

This paper contains 2 sections, 4 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

Let $A=M_n(D)$ be the algebra of $n\times n$ square matrices over a division ring $D$ with characteristics different from $2$. Let $f,g: A \rightarrow A$ be additive functions, satisfying identity moraes for every invertible $x$. Then $f(x)=g(x)=0$, for all $x\in A$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Example 2.1
  • Theorem 2.3
  • proof