Decomposition theorems of modules over commutative rings

Abstract

[en] The Fundamental Theorem of Finitely Generated Abelian Groups is a very important result that allows us to describe explicitly the finitely generated abelian groups. This theorem can naturally be generalized to finitely generated modules over principal ideal domains. The difficulty arises when we try to extend it to other types of rings. In this work we prove a generalization of this, the decomposition theorem of finitely generated modules over Dedekind domains. We also explore other similar decomposition and we characterize the rings such that all their modules have a decomposition of simple and cyclic modules.