2-Modules and the Representation of 2-Rings

TL;DR

This paper develops a 2-dimensional algebraic framework for modules and 2-rings, establishing foundational definitions and showing that modules over a 2-ring form a 2-Abelian category.

Contribution

It introduces the concepts of 2-modules and the representation of 2-rings, extending classical module theory into a higher-dimensional setting.

Findings

Defined 2-modules and 2-ring representations

Proved modules over a 2-ring form a 2-Abelian category

Extended classical algebraic structures into 2-dimensional algebra

Abstract

In this paper, we develop 2-dimensional algebraic theory which closely follows the classical theory of modules. The main results are giving definitions of 2-module and the representation of 2-ring. Moreover, for a 2-ring $\cR$, we prove that its modules form a 2-Abelian category.