Factorization Theory and Decompositions of Modules

Let R be a commutative ring with identity. It often happens that M-1 circle plus center dot center dot center dot circle plus M-s congruent to N-1 circle plus center dot center dot center dot circle plus N-t for indecomposable R-modules M-1, . . . , M-s and N-1, . . . , N-t with s not equal t. This behavior can be captured by studying the commutative monoid {[M] vertical bar M is an R-module} of isomorphism classes of R-modules with operation given by [M] + [N] = [M circle plus 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.