Our object in this chapter is to prove the fundamental structure theorem for finitely generated modules over a principal ideal domain. This theorem provides a decomposition of such a module into a direct sum of cyclic modules. In particular, it gives an independent proof of the fundamental theorem of finitely generated abelian groups proved earlier in Chapter 8. Applications of this theorem to linear algebra are given in Sections 4 and 5.

**1.1 Theorem.** *Let R be a principal ideal domain, and let M be any finitely generated R-module. Then*

*a direct sum of cyclic modules, ...*

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