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On Jordan superderivations and Jordan super-biderivations of trivial extensions and triangular matrix rings

Hassan Cheraghpour, Madineh Jafari

Abstract

Triangular matrix rings are example of trivial extensions. In this article we describe the Jordan superderivations of the trivial extensions and upper triangular matrix rings. We deduce then that any Jordan superderivation of an upper triangular matrix ring, under some conditions, is a derivation, and any Jordan super-biderivation of a trivial extension, and hence an upper triangular matrix ring, is a Jordan biderivation.

On Jordan superderivations and Jordan super-biderivations of trivial extensions and triangular matrix rings

Abstract

Triangular matrix rings are example of trivial extensions. In this article we describe the Jordan superderivations of the trivial extensions and upper triangular matrix rings. We deduce then that any Jordan superderivation of an upper triangular matrix ring, under some conditions, is a derivation, and any Jordan super-biderivation of a trivial extension, and hence an upper triangular matrix ring, is a Jordan biderivation.
Paper Structure (2 sections, 6 theorems, 42 equations)

This paper contains 2 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Main results and proofs

Key Result

Theorem 2.1

Let $d_{0}$ and $d_{1}$ be, respectively, Jordan superderivations of degree $0$ and $1$ of the trivial extension $T(R,M)$. Then there exists such that $d_{0}$ and $d_{1}$ can be expressed as Hence, every Jordan superderivation of $T(R,M)$ is of the form where $\delta$, $f$, $g$ and $\gamma$ are given above.

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Theorem 2.6
  • ...and 2 more