Theorem 1 (Azumaya)

Tag 00RX, permanent. Cite it as [Bourbaki, Tag 00RX].

Algebra, VIII, p. 32, § 2. The Structure of Modules of Finite Length

Let A be a ring, L a primordial A-module, and M a semiprimordial A-module. There exists a unique cardinal, denoted by [M : L], with the following property:

For every decomposition $M =\bigoplus_{i\in I}M_i$ of M as a direct sum of primordial modules, the set of indices $i\in I$ such that $M_i$ is isomorphic to L has cardinal [M : L].

The proof is based on the following four lemmas.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Proposition 6, § 10
Lemma 4, § 2
Theorem 2 (Krull–Remak–Schmidt), § 2
lie-i-s3-prop-13 not in this corpus
lie-v-a0-prop-1 not in this corpus

Cites

Nothing.