Factorization Theory and Decompositions of Modules

Abstract

Let R be a commutative ring with identity. It often happens that M1 ⊕ ⋯ ⊕ Ms ≅ N1 ⊕ ⋯ ⊕ Nt for indecomposable R-modules M1, …, Ms and N1, …, Nt with s ≠ t. This behavior can be captured by studying the commutative monoid {[M] ❘ M is an R-module} of isomorphism classes of R-modules with operation given by [M] + [N] = [M ⊕ N]. In this mostly self-contained exposition, we introduce the reader to the interplay between the the study of direct-sum decompositions of modules and the study of factorizations in integral domains.