Free Hom-groups, Hom-rings and Semisimple modules

TL;DR

This paper introduces Hom-type generalizations of rings and groups, exploring their properties, modules, and simple structures, extending classical algebraic concepts to this new framework.

Contribution

It develops foundational properties of Hom-rings and Hom-groups, including constructions, module categories, and simple objects, extending classical algebraic results.

Findings

Characterization of simple modules over Hom-rings

Construction of free regular Hom-groups using Super-Leaf weighted trees

Extension of classical group results to Hom-groups

Abstract

The purpose of this paper is to introduce and study a Hom-type generalization of rings. We provide their basic properties and and some key constructions. Furthermore, we consider modules over Hom-rings and characterize the category of simple modules and simple Hom-rings. In addition, we extend some classical results and concepts of groups to Hom-groups. We construct free regular Hom-group using Super-Leaf weighted trees and discuss Normal Hom-subgroups, abelianization of regular Hom-group, universality of tensor product of Hom-groups and simple Hom-groups.