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Additivity of multiplicative (generalized) maps over rings

Sk Aziz, Arindam Ghosh, Om Prakash

Abstract

In this paper, we mainly prove some results on the additivity of maps over rings under certain conditions. First, we discuss a special case of MARTINDALE III's theorem of \cite{1969M} as a bijective map $\varphi$ over a ring $R$ with a non-trivial idempotent satisfying $\varphi(ab)=\varphi(a)\varphi(b)$ for all $a, b\in R$, is additive. Then we prove that a map $D$ on $R$ satisfying $D(ab)=D(a)b+\varphi(a) D(b)$ for all $a,b\in R$, where $\varphi$ is the map mentioned above, is additive. Finally, we establish that if a map $g$ over $R$ satisfies $g(ab)=g(a)b+\varphi(a)D(b),$ for all $a,b\in R$ and the maps $\varphi$ and $D$ are mentioned above, then $g$ is additive.

Additivity of multiplicative (generalized) maps over rings

Abstract

In this paper, we mainly prove some results on the additivity of maps over rings under certain conditions. First, we discuss a special case of MARTINDALE III's theorem of \cite{1969M} as a bijective map over a ring with a non-trivial idempotent satisfying for all , is additive. Then we prove that a map on satisfying for all , where is the map mentioned above, is additive. Finally, we establish that if a map over satisfies for all and the maps and are mentioned above, then is additive.
Paper Structure (4 sections, 21 theorems, 92 equations)

This paper contains 4 sections, 21 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Multiplicative Automorphism
  3. Multiplicative Skew Derivation
  4. Multiplicative Generalized Skew Derivation

Key Result

Theorem 2.1

If $\varphi$ is a multiplicative automorphism on $R$, then $\varphi$ is additive. Moreover, $\varphi$ is an automorphism on $R$.

Theorems & Definitions (42)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 32 more