The Relation between Ann-Categories and Ring Categories

TL;DR

This paper proves that the axioms defining a ring category can be derived from the axioms of an Ann-category, unifying two categorical approaches to ring structures.

Contribution

It demonstrates that ring category axioms are consequences of Ann-category axioms, clarifying the relationship between these two categorical frameworks.

Findings

Ring category axioms follow from Ann-category axioms

Unification of categorical definitions of rings

Simplification of ring category axiomatization

Abstract

There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom of the definition of a {\it ring category}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an {\it Ann-category}.