Proposition 8

Tag 074E, permanent. Cite it as [Bourbaki, Tag 074E].

Algebra, chapter VII, § 4. Finitely generated modules over a principal ideal domain

— Let $A$ be a principal ideal domain and let $M$ be a finitely generated $A$ -module.

a) $M$ is indecomposable if and only if it is isomorphic to $A$ or to a module of the form $A/(p^n)$ , where $p$ is an irreducible element of $A$ and $n > 0$ is an integer.

b) $M$ is a direct sum of a finite family of indecomposable submodules.

Part b) of the above proposition can be made more precise as follows:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 9, § 4
Example, § 2
Example 2, § 3

Cites

Nothing.