# Characterizations of derivations

## TL;DR

This paper characterizes derivations using a single functional equation, explores the solvability of related systems, and investigates conditions under which additive functions are sums of derivations and linear functions.

## Contribution

It introduces a unified functional equation to characterize derivations and analyzes the additive solutions and their linear independence, extending understanding of derivations in rings.

## Key findings

- Derivations can be characterized by a specific functional equation under certain ring conditions.
- The additive solutions of the functional system are linearly dependent or independent based on the equations.
- Additive functions satisfying certain regularity conditions can be expressed as sums of derivations and linear functions.

## Abstract

The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 we collect all the definitions and results regarding derivations that are essential while studying this area. In Chapter 3 we intend to show that derivations can be characterized by one single functional equation. More exactly, we study here the following problem. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $\lambda, \mu\in Q\setminus\left\{0\right\}$ be arbitrary, $f\colon P\rightarrow Q$ be a function and consider the equation \[ \lambda\left[f(x+y)-f(x)-f(y)\right]+ \mu\left[f(xy)-xf(y)-yf(x)\right]=0 \quad \left(x, y\in P\right). \] In this chapter it will be proved that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation. Chapter 4 is devoted to the additive solvability of a system of functional equations. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n} \colon\mathbb{R}\to\mathbb{R}$ of the above system of equations is characterized. Finally, the closing chapter deals with the following problem. Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for the additive function $f\colon \mathbb{R}\to \mathbb{R}$, the mapping \[ \varphi(x)=f\left(\xi(x)\right)-\xi'(x)f(x) \] fulfills some regularity condition on its domain. Is it true that in such a case $f$ is a sum of a derivation and a linear function?