On the independence of conditions in the linear mapping definition (revised)

TL;DR

This paper investigates the independence of additivity and homogeneity conditions in the definition of linear mappings over various fields, revealing their dependence only over prime fields and independence over non-prime fields.

Contribution

It provides a comprehensive analysis of the dependence relations between linearity conditions across different scalar fields, extending previous work to the general case.

Findings

Over prime fields, additivity implies homogeneity.

Over non-prime fields, additivity and homogeneity are independent.

The problem posed in the previous version is fully resolved.

Abstract

We study the (in)dependence of additivity and homogeneity conditions in the definition of linear mappings between vector spaces over the same scalar field. Unlike other works on the subject, dealing with particular fields like real or complex numbers, or with particular mappings like continuous or measurable, we consider the general case. This enables us to obtain complete picture. Namely, for the prime field, and only in this case, the conditions are dependent (additivity implies homogeneity). For the non-prime field they are independent: neither of conditions implies the other. Thus, the problem posed in the previous version of the paper, is solved here.