An abstract factorization theorem and some applications

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