Corollary

Tag 073P, permanent. Cite it as [Bourbaki, Tag 073P].

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

— In the notation of Prop. 2, if $a_n \neq A$ , and if $E$ is also isomorphic to the direct sum of $m$ cyclic modules $A / a'_j$ with $a'_1 \subset a'_2 \subset \ldots \subset a'_m \neq A$ , then $m = n$ and $a_k = a'_k$ for $1 \leq k \leq n$ (uniqueness of the $a_k *$ ).

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

Cited by

Proposition 9, § 4
Theorem 1, § 4

Cites

Prop. 2