  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").
