An Abstract Factorization Theorem and Some Applications

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").