An Abstract Factorization Theorem and Some Applications

TL;DR

This paper introduces an abstract factorization theorem by integrating monoid and preorder theories, extending classical results to broader algebraic structures and applying them to modules and additive number theory.

Contribution

It generalizes classical factorization theorems to Dedekind-finite monoids and provides new proofs and applications in module theory and additive combinatorics.

Findings

Generalization of atomic factorization theorems to Dedekind-finite monoids

A monoid-theoretic proof that modules of finite uniform dimension decompose into indecomposables

Enhanced understanding of factorizations in algebraic and combinatorial structures

Abstract

We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular, we obtain a generalization, from cancellative to Dedekind-finite (commutative or non-commutative) monoids, of a classical theorem on "atomic factorizations" that traces back to the work of P.M. Cohn in the 1960s; recover a theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings; improve on a theorem of A.A. Antoniou and the author that characterizes atomicity in certain "monoids of sets" naturally arising from additive number theory and arithmetic combinatorics; and give a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring $R$ is a direct sum of finitely many indecomposable modules (this is in fact a special case of a more general decomposition theorem for the objects of certain categories with finite products, where the indecomposable $R$-modules are characterized as the atoms of a suitable "monoid of modules").